Octave 3.8, jcobi/1

Percentage Accurate: 75.1% → 99.7%

Time: 11.0s

Alternatives: 14

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
     (/
      (fma 0.5 (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha)) (+ beta 1.0))
      alpha)
     (* 0.5 (+ (/ beta t_0) (- 1.0 (/ alpha t_0)))))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = fma(0.5, ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)), (beta + 1.0)) / alpha;
	} else {
		tmp = 0.5 * ((beta / t_0) + (1.0 - (alpha / t_0)));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(fma(0.5, Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)), Float64(beta + 1.0)) / alpha);
	else
		tmp = Float64(0.5 * Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(N[(0.5 * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(0.5 * N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.98999999999999999

   1. Initial program 6.7%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-sub N/A

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

      9. sub-neg N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. div-inv N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;0.5 \cdot \left(t\_1 + \frac{\beta - -2}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_1 + \left(1 - \frac{\alpha}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
     (* 0.5 (+ t_1 (/ (- beta -2.0) alpha)))
     (* 0.5 (+ t_1 (- 1.0 (/ alpha t_0)))))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = 0.5 * (t_1 + ((beta - -2.0) / alpha));
	} else {
		tmp = 0.5 * (t_1 + (1.0 - (alpha / t_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) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99d0)) then
        tmp = 0.5d0 * (t_1 + ((beta - (-2.0d0)) / alpha))
    else
        tmp = 0.5d0 * (t_1 + (1.0d0 - (alpha / t_0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = 0.5 * (t_1 + ((beta - -2.0) / alpha));
	} else {
		tmp = 0.5 * (t_1 + (1.0 - (alpha / t_0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99:
		tmp = 0.5 * (t_1 + ((beta - -2.0) / alpha))
	else:
		tmp = 0.5 * (t_1 + (1.0 - (alpha / t_0)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(0.5 * Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)));
	else
		tmp = Float64(0.5 * Float64(t_1 + Float64(1.0 - Float64(alpha / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99)
		tmp = 0.5 * (t_1 + ((beta - -2.0) / alpha));
	else
		tmp = 0.5 * (t_1 + (1.0 - (alpha / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(0.5 * N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$1 + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.98999999999999999

   1. Initial program 6.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 N/A

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

   4. Applied rewrites 8.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-sub N/A

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

      9. sub-neg N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. div-inv N/A

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

   6. Applied rewrites 8.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 8.0

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in alpha around inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-sub N/A

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

      9. sub-neg N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. div-inv N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha \cdot \alpha, 0.125, \mathsf{fma}\left(\alpha, -0.25, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 1e-6)
       (fma (* alpha alpha) 0.125 (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 = (beta + 1.0) / alpha;
	} else if (t_0 <= 1e-6) {
		tmp = fma((alpha * alpha), 0.125, 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(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 1e-6)
		tmp = fma(Float64(alpha * alpha), 0.125, 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[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[(N[(alpha * alpha), $MachinePrecision] * 0.125 + N[(alpha * -0.25 + 0.5), $MachinePrecision]), $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 8.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999955e-7

   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. lower-+.f64 N/A

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

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

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

      13. metadata-eval 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   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. *-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. lower-fma.f64 98.2

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

   8. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 98.2

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

   10. Applied rewrites 98.2%

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

   if 9.99999999999999955e-7 < (/.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. Applied rewrites 97.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha \cdot \alpha, 0.125, \mathsf{fma}\left(\alpha, -0.25, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{t\_0} + \frac{\beta - -2}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
     (* 0.5 (+ (/ beta t_0) (/ (- beta -2.0) alpha)))
     (fma (/ (- beta alpha) t_0) 0.5 0.5))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = 0.5 * ((beta / t_0) + ((beta - -2.0) / alpha));
	} else {
		tmp = fma(((beta - alpha) / t_0), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(0.5 * Float64(Float64(beta / t_0) + Float64(Float64(beta - -2.0) / alpha)));
	else
		tmp = fma(Float64(Float64(beta - alpha) / t_0), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(0.5 * N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $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))) < -0.98999999999999999

   1. Initial program 6.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 N/A

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

   4. Applied rewrites 8.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-sub N/A

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

      9. sub-neg N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. div-inv N/A

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

   6. Applied rewrites 8.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 8.0

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in alpha around inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 99.3

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

   11. Applied rewrites 99.3%

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

   if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;0.5 \cdot \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 1e-6) (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 = (beta + 1.0) / alpha;
	} else if (t_0 <= 1e-6) {
		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(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 1e-6)
		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[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[(alpha * N[(alpha * 0.125 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999955e-7

   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. lower-+.f64 N/A

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

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

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

      13. metadata-eval 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   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. *-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. lower-fma.f64 98.2

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

   8. Applied rewrites 98.2%

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

   if 9.99999999999999955e-7 < (/.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. Applied rewrites 97.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 1e-6) (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 <= 1e-6) {
		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 <= 1e-6)
		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, 1e-6], N[(alpha * N[(alpha * 0.125 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.0%

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

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

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

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

      13. metadata-eval 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999955e-7

   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. lower-+.f64 N/A

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

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

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

      13. metadata-eval 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   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. *-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. lower-fma.f64 98.2

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

   8. Applied rewrites 98.2%

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

   if 9.99999999999999955e-7 < (/.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. Applied rewrites 97.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 1e-6) (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 <= 1e-6) {
		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 <= 1e-6)
		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, 1e-6], 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 8.0%

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

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

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

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

      13. metadata-eval 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999955e-7

   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. lower-+.f64 N/A

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

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

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

      13. metadata-eval 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   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. lower-fma.f64 98.0

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

   8. Applied rewrites 98.0%

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

   if 9.99999999999999955e-7 < (/.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. Applied rewrites 97.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
   (/ (+ beta 1.0) alpha)
   (fma (/ (- beta alpha) (+ beta (+ alpha 2.0))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (beta + (alpha + 2.0))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.98999999999999999

   1. Initial program 6.7%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
   (/ (+ 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)) <= -0.99) {
		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)) <= -0.99)
		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], -0.99], 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))) < -0.98999999999999999

   1. Initial program 6.7%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-inv N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% 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 8.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Applied rewrites 99.8%

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

4. Final simplification 99.4%

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

Alternative 11: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.8%

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

Alternative 12: 71.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 1e-6) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 1e-6) {
		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)) <= 1d-6) 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)) <= 1e-6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 1e-6:
		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)) <= 1e-6)
		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)) <= 1e-6)
		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], 1e-6], 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))) < 9.99999999999999955e-7

   1. Initial program 66.5%

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

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

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

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

      13. metadata-eval 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 63.3%

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

   if 9.99999999999999955e-7 < (/.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. Applied rewrites 97.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.6% 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 76.3%

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

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

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

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

   13. metadata-eval 50.5

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 50.3%

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

Alternative 14: 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 76.3%

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

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

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

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

   13. metadata-eval 50.5

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 3.7%

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

   1. metadata-eval 3.7

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

10. Applied rewrites 3.7%

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

Reproduce

herbie shell --seed 2024216 
(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))