Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.9%

Time: 10.9s

Alternatives: 14

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
   (/ (- beta (fma (+ beta 2.0) (/ (+ beta 1.0) alpha) -1.0)) alpha)
   (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = (beta - fma((beta + 2.0), ((beta + 1.0) / alpha), -1.0)) / alpha;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 1.0) / 2.0;
	}
	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 - fma(Float64(beta + 2.0), Float64(Float64(beta + 1.0) / alpha), -1.0)) / alpha);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 1.0) / 2.0);
	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[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in alpha around inf

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

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

   6. Taylor expanded in alpha around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

   8. Simplified 100.0%

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

   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

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

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

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 0.5)
       (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 <= 0.5) {
		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 <= 0.5)
		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, 0.5], 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 8.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 98.0

          \[\leadsto 0.5 + \frac{\alpha}{\color{blue}{-2 - \alpha}} \cdot 0.5 \]

   7. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 + \frac{\alpha}{-2 - \alpha} \cdot 0.5} \]

   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 97.7

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

   10. Simplified 97.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around -inf

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

      1. +-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. sub-neg N/A

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

      7. *-lft-identity N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{\alpha}{\beta}} \]
   10. 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 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 98.3%

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

Alternative 3: 97.4% accurate, 0.5× speedup

Math

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

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 0.5)
       (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 <= 0.5) {
		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 <= 0.5)
		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, 0.5], 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 8.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 98.0

          \[\leadsto 0.5 + \frac{\alpha}{\color{blue}{-2 - \alpha}} \cdot 0.5 \]

   7. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 + \frac{\alpha}{-2 - \alpha} \cdot 0.5} \]

   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 97.5

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

   10. Simplified 97.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around -inf

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

      1. +-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. sub-neg N/A

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

      7. *-lft-identity N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{\alpha}{\beta}} \]
   10. 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 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 98.2%

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

Alternative 4: 92.2% accurate, 0.5× speedup

Math

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

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 0.5)
       (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 <= 0.5) {
		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 <= 0.5)
		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, 0.5], 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 8.7%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

      14. metadata-eval 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 84.0%

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

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 98.0

          \[\leadsto 0.5 + \frac{\alpha}{\color{blue}{-2 - \alpha}} \cdot 0.5 \]

   7. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 + \frac{\alpha}{-2 - \alpha} \cdot 0.5} \]

   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 97.5

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

   10. Simplified 97.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around -inf

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

      1. +-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. sub-neg N/A

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

      7. *-lft-identity N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{\alpha}{\beta}} \]
   10. 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 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 94.2%

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

Alternative 5: 92.1% accurate, 0.5× speedup

Math

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

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 0.5) (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 <= 0.5) {
		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 <= 0.5)
		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, 0.5], 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 8.7%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

      14. metadata-eval 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 84.0%

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

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 98.0

          \[\leadsto 0.5 + \frac{\alpha}{\color{blue}{-2 - \alpha}} \cdot 0.5 \]

   7. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 + \frac{\alpha}{-2 - \alpha} \cdot 0.5} \]

   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 97.5

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

   10. Simplified 97.5%

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

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

   1. Initial program 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 99.5%

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

4. Final simplification 94.1%

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

Alternative 6: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
   (/ (- beta (fma (+ beta 2.0) (/ beta alpha) -1.0)) alpha)
   (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = (beta - fma((beta + 2.0), (beta / alpha), -1.0)) / alpha;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 1.0) / 2.0;
	}
	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 - fma(Float64(beta + 2.0), Float64(beta / alpha), -1.0)) / alpha);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 1.0) / 2.0);
	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[(beta + 2.0), $MachinePrecision] * N[(beta / alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in alpha around inf

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

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

   6. Taylor expanded in alpha around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 99.0

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

   11. Simplified 99.0%

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

   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

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

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

Alternative 7: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

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 - fma((beta + 2.0), (beta / alpha), -1.0)) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	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 <= -0.999995)
		tmp = Float64(Float64(beta - fma(Float64(beta + 2.0), Float64(beta / alpha), -1.0)) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	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, -0.999995], N[(N[(beta - N[(N[(beta + 2.0), $MachinePrecision] * N[(beta / alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in alpha around inf

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

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

   6. Taylor expanded in alpha around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in beta around inf

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

       1. /-lowering-/.f64 99.0

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

   11. Simplified 99.0%

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

   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. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 8: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999995d0)) then
        tmp = ((-0.5d0) * (((-2.0d0) - beta) - beta)) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999995:
		tmp = (-0.5 * ((-2.0 - beta) - beta)) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	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 <= -0.999995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(-2.0 - beta) - beta)) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999995)
		tmp = (-0.5 * ((-2.0 - beta) - beta)) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = 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[(-0.5 * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

      14. metadata-eval 98.7

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

   5. Simplified 98.7%

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

   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. Recombined 2 regimes into one program.

4. Final simplification 99.5%

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

Alternative 9: 92.0% accurate, 0.6× speedup

Math

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

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 0.5) (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 <= 0.5) {
		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 <= 0.5)
		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, 0.5], 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 8.7%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

      14. metadata-eval 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 84.0%

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

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

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 98.0

          \[\leadsto 0.5 + \frac{\alpha}{\color{blue}{-2 - \alpha}} \cdot 0.5 \]

   7. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 + \frac{\alpha}{-2 - \alpha} \cdot 0.5} \]

   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 96.8

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

   10. Simplified 96.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 99.5%

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

4. Final simplification 93.7%

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

Alternative 10: 98.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
   (/ (* -0.5 (- (- -2.0 beta) beta)) 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 = (-0.5 * ((-2.0 - beta) - beta)) / 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(-0.5 * Float64(Float64(-2.0 - beta) - beta)) / 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[(-0.5 * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $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 8.7%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

      14. metadata-eval 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

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

   3. 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 97.9

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

   5. Simplified 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\beta}{\beta + 2}, 0.5\right)\\ \end{array} \]
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 8.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 97.9

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

   5. Simplified 97.9%

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

   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 97.9

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

   5. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 12: 71.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= 0.5d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 62.5

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

   5. Simplified 62.5%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 61.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 99.5%

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

4. Final simplification 70.2%

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

Alternative 13: 72.3% 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 68.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 65.2

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

   5. Simplified 65.2%

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

   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 64.1

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

   8. Simplified 64.1%

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

   if 2 < beta

   1. Initial program 88.2%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 87.3%

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

Alternative 14: 49.2% 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 73.6%

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

3. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. distribute-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 71.2

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

5. Simplified 71.2%

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

6. Taylor expanded in beta around 0

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

8. Simplified 51.3%

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

Reproduce

herbie shell --seed 2024204 
(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))