Octave 3.8, jcobi/1

Percentage Accurate: 74.2% → 99.3%

Time: 10.2s

Alternatives: 12

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((beta - alpha), (0.5 / (beta + (alpha + 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)) <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(beta - alpha), Float64(0.5 / Float64(beta + Float64(alpha + 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], -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-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 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.0625, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

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 0.2)
       (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 <= 0.2) {
		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 <= 0.2)
		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, 0.2], 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 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-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 98.1

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

   5. Simplified 98.1%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\alpha \cdot \left(\frac{1}{8} + \frac{-1}{16} \cdot \alpha\right) - \frac{1}{4}\right)} \]
   7. 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 96.6

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

   8. Simplified 96.6%

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

   if 0.20000000000000001 < (/.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 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   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 95.9

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

   11. Simplified 95.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.0625, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

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 0.2)
       (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 <= 0.2) {
		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 <= 0.2)
		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, 0.2], 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 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-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 98.1

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

   5. Simplified 98.1%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   7. 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 96.3

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

   8. Simplified 96.3%

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

   if 0.20000000000000001 < (/.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 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   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 95.9

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

   11. Simplified 95.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

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 0.2)
       (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 <= 0.2) {
		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 <= 0.2)
		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, 0.2], 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 7.3%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 6.1

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

   5. Simplified 6.1%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.6

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

   8. Simplified 81.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   7. 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 96.3

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

   8. Simplified 96.3%

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

   if 0.20000000000000001 < (/.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 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   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 95.9

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

   11. Simplified 95.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.2)
       (fma alpha (fma alpha 0.125 -0.25) 0.5)
       (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.2) {
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.2)
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(alpha * N[(alpha * 0.125 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.3%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 6.1

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

   5. Simplified 6.1%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.6

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

   8. Simplified 81.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   7. 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 96.3

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

   8. Simplified 96.3%

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

   if 0.20000000000000001 < (/.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 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in alpha around 0

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. metadata-eval N/A

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

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

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

      5. /-lowering-/.f64 95.7

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

   8. Simplified 95.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.2) (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 <= 0.2) {
		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 <= 0.2)
		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, 0.2], 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 7.3%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 6.1

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

   5. Simplified 6.1%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.6

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

   8. Simplified 81.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   7. 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 96.3

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

   8. Simplified 96.3%

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

   if 0.20000000000000001 < (/.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 94.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.2) (fma alpha -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 <= 0.2) {
		tmp = fma(alpha, -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 <= 0.2)
		tmp = fma(alpha, -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, 0.2], N[(alpha * -0.25 + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.3%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 6.1

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

   5. Simplified 6.1%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.6

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

   8. Simplified 81.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 95.7

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

   8. Simplified 95.7%

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

   if 0.20000000000000001 < (/.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 94.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
   (/ (+ beta 1.0) alpha)
   (fma (/ (- beta alpha) (+ beta 2.0)) 0.5 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(((beta - alpha) / (beta + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-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 98.1

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

   5. Simplified 98.1%

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

   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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

   7. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 9: 98.3% 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 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-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 98.1

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

   5. Simplified 98.1%

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

   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 97.6

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

   5. Simplified 97.6%

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

4. Final simplification 97.7%

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

Alternative 10: 71.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 0.2) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2) {
		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)) <= 0.2d0) 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)) <= 0.2) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2:
		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)) <= 0.2)
		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)) <= 0.2)
		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], 0.2], 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))) < 0.20000000000000001

   1. Initial program 64.6%

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

   3. Taylor expanded in beta around 0

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

      1. 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-lft-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 60.7%

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

   if 0.20000000000000001 < (/.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 94.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.1% 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 75.4%

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

3. Taylor expanded in beta around 0

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

   1. 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-lft-in N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. distribute-neg-frac2 N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. distribute-lft-in N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

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

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

   13. metadata-eval 48.2

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

5. Simplified 48.2%

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

6. Taylor expanded in alpha around 0

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

8. Simplified 47.9%

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

Alternative 12: 3.7% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.0

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in beta around 0

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

   1. 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-lft-in N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. distribute-neg-frac2 N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. distribute-lft-in N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

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

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

   13. metadata-eval 48.2

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

5. Simplified 48.2%

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

6. Taylor expanded in alpha around inf

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

8. Simplified 3.6%

   \[\leadsto 0.5 + \color{blue}{-0.5} \]
9. Step-by-step derivation

   1. metadata-eval 3.6

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

10. Applied egg-rr 3.6%

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

Reproduce

herbie shell --seed 2024199 
(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))