Octave 3.8, jcobi/1

Percentage Accurate: 74.1% → 99.8%

Time: 8.5s

Alternatives: 12

Speedup: 0.6×

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: 74.1% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ t_1 := \frac{\beta + 2}{\alpha}\\ t_2 := \left(\beta + 2\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(2, \beta, 2\right) + t\_2 \cdot \left(t\_1 + -1\right)\right) + \frac{\beta}{\alpha} \cdot \left(t\_2 - \left(\beta + 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0)))
        (t_1 (/ (+ beta 2.0) alpha))
        (t_2 (* (+ beta 2.0) t_1)))
   (if (<= t_0 -0.95)
     (/
      (/
       (+
        (+ (fma 2.0 beta 2.0) (* t_2 (+ t_1 -1.0)))
        (* (/ beta alpha) (- t_2 (+ beta 2.0))))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double t_1 = (beta + 2.0) / alpha;
	double t_2 = (beta + 2.0) * t_1;
	double tmp;
	if (t_0 <= -0.95) {
		tmp = (((fma(2.0, beta, 2.0) + (t_2 * (t_1 + -1.0))) + ((beta / alpha) * (t_2 - (beta + 2.0)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	t_1 = Float64(Float64(beta + 2.0) / alpha)
	t_2 = Float64(Float64(beta + 2.0) * t_1)
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(Float64(Float64(fma(2.0, beta, 2.0) + Float64(t_2 * Float64(t_1 + -1.0))) + Float64(Float64(beta / alpha) * Float64(t_2 - Float64(beta + 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] + N[(t$95$2 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / alpha), $MachinePrecision] * N[(t$95$2 - N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.94999999999999996

   1. Initial program 7.7%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. associate-*r/ N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-in N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 N/A

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

      21. metadata-eval 7.7

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

   5. Simplified 7.7%

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

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(2 + \left(-1 \cdot \frac{{\left(2 + \beta\right)}^{2}}{\alpha} + \left(2 \cdot \beta + \frac{{\left(2 + \beta\right)}^{3}}{{\alpha}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\beta \cdot {\left(2 + \beta\right)}^{2}}{{\alpha}^{2}} + \frac{\beta \cdot \left(2 + \beta\right)}{\alpha}\right)}{\alpha}}}{2} \]

   8. Simplified 99.9%

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

   if -0.94999999999999996 < (/.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. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(2, \beta, 2\right) + \left(\left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right) \cdot \left(\frac{\beta + 2}{\alpha} + -1\right)\right) + \frac{\beta}{\alpha} \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha} - \left(\beta + 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta + 2, \frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\alpha}, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.95)
     (/
      (/
       (fma (+ beta 2.0) (/ (fma beta -2.0 -2.0) alpha) (fma 2.0 beta 2.0))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = (fma((beta + 2.0), (fma(beta, -2.0, -2.0) / alpha), fma(2.0, beta, 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(fma(Float64(beta + 2.0), Float64(fma(beta, -2.0, -2.0) / alpha), fma(2.0, beta, 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta * -2.0 + -2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.94999999999999996

   1. Initial program 7.7%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 99.7%

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

   if -0.94999999999999996 < (/.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. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta + 2, \frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\alpha}, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-0.0625, \alpha, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.004)
       (fma alpha (fma alpha (fma -0.0625 alpha 0.125) -0.25) 0.5)
       (+ (/ -1.0 beta) 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 = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.004) {
		tmp = fma(alpha, fma(alpha, fma(-0.0625, alpha, 0.125), -0.25), 0.5);
	} else {
		tmp = (-1.0 / beta) + 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(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.004)
		tmp = fma(alpha, fma(alpha, fma(-0.0625, alpha, 0.125), -0.25), 0.5);
	else
		tmp = Float64(Float64(-1.0 / beta) + 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[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(alpha * N[(alpha * N[(-0.0625 * alpha + 0.125), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in beta around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 97.4

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

   8. Simplified 97.4%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0040000000000000001

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.8

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

   5. Simplified 98.8%

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

   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. lower-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. lower-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. lower-fma.f64 97.8

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

   8. Simplified 97.8%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-0.0625, \alpha, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.004)
       (fma alpha (fma 0.125 alpha -0.25) 0.5)
       (+ (/ -1.0 beta) 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 = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.004) {
		tmp = fma(alpha, fma(0.125, alpha, -0.25), 0.5);
	} else {
		tmp = (-1.0 / beta) + 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(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.004)
		tmp = fma(alpha, fma(0.125, alpha, -0.25), 0.5);
	else
		tmp = Float64(Float64(-1.0 / beta) + 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[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(alpha * N[(0.125 * alpha + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in beta around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 97.4

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

   8. Simplified 97.4%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0040000000000000001

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.8

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

   5. Simplified 98.8%

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

   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. lower-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. metadata-eval N/A

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

      5. lower-fma.f64 97.3

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

   8. Simplified 97.3%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \end{array} \]

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 0.004)
       (fma alpha (fma 0.125 alpha -0.25) 0.5)
       (+ (/ -1.0 beta) 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 <= 0.004) {
		tmp = fma(alpha, fma(0.125, alpha, -0.25), 0.5);
	} else {
		tmp = (-1.0 / beta) + 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 <= 0.004)
		tmp = fma(alpha, fma(0.125, alpha, -0.25), 0.5);
	else
		tmp = Float64(Float64(-1.0 / beta) + 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, 0.004], N[(alpha * N[(0.125 * alpha + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 7.2

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

   5. Simplified 7.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 80.6

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

   8. Simplified 80.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0040000000000000001

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.8

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

   5. Simplified 98.8%

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

   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. lower-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. metadata-eval N/A

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

      5. lower-fma.f64 97.3

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

   8. Simplified 97.3%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.004) (fma alpha (fma 0.125 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 <= 0.004) {
		tmp = fma(alpha, fma(0.125, 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 <= 0.004)
		tmp = fma(alpha, fma(0.125, 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, 0.004], N[(alpha * N[(0.125 * alpha + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 7.2

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

   5. Simplified 7.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 80.6

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

   8. Simplified 80.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0040000000000000001

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.8

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

   5. Simplified 98.8%

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

   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. lower-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. metadata-eval N/A

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

      5. lower-fma.f64 97.3

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

   8. Simplified 97.3%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 99.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(0.125, \alpha, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999998) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in beta around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 99.1

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

   8. Simplified 99.1%

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

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

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 8: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.004) (fma -0.25 alpha 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.004) {
		tmp = fma(-0.25, alpha, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.004)
		tmp = fma(-0.25, alpha, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(-0.25 * alpha + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 7.2

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

   5. Simplified 7.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 80.6

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

   8. Simplified 80.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0040000000000000001

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.8

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

   5. Simplified 98.8%

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

   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. lower-fma.f64 96.8

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

   8. Simplified 96.8%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

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 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.8%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in beta around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 97.4

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

   8. Simplified 97.4%

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

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 98.1%

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

Alternative 10: 70.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 0.004) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.004) {
		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)) <= 0.004d0) 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)) <= 0.004) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.004:
		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)) <= 0.004)
		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)) <= 0.004)
		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], 0.004], 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))) < 0.0040000000000000001

   1. Initial program 61.8%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 58.9

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

   5. Simplified 58.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 58.2%

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

   if 0.0040000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 99.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.004:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 65.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.5

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

   5. Simplified 62.5%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 62.2

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

   8. Simplified 62.2%

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

   if 2 < beta

   1. Initial program 87.9%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.6

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

   5. Simplified 86.6%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 86.2%

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

Alternative 12: 48.7% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 70.4

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

5. Simplified 70.4%

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

6. Taylor expanded in beta around 0

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

8. Simplified 47.1%

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

Reproduce

herbie shell --seed 2024215 
(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))