Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.9%

Time: 9.5s

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: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(-1.0 / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.5%

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

   3. Taylor expanded in alpha around inf

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

      1. 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. Simplified 99.8%

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

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

   1. Initial program 99.9%

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

      1. 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

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.02)
       (fma beta (fma beta (fma beta 0.0625 -0.125) 0.25) 0.5)
       (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.02) {
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 0.5);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.02)
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 0.5);
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(beta * N[(beta * N[(beta * 0.0625 + -0.125), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 97.4

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

   5. Simplified 97.4%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0200000000000000004

   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 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 97.8

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

   8. Simplified 97.8%

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

   if 0.0200000000000000004 < (/.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 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.7

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

   8. Simplified 97.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))