Octave 3.8, jcobi/1

Percentage Accurate: 63.3% → 81.7%

Time: 3.0s

Alternatives: 9

Speedup: 0.9×

• Could not converge on a ground truth

  1. alpha = 1.7533746046869366e+195
  2. beta = -1.7030234586554808e+258

Specification

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

Math

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

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)
use fmin_fmax_functions
    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: 63.3% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq -5200000:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (if (<= beta -5200000.0)
  (- -0.5 -0.5)
  (fma (/ (- alpha beta) (- -2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -5200000.0) {
		tmp = -0.5 - -0.5;
	} else {
		tmp = fma(((alpha - beta) / (-2.0 - (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -5200000.0)
		tmp = Float64(-0.5 - -0.5);
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -5200000.0], N[(-0.5 - -0.5), $MachinePrecision], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < -5.2e6

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if -5.2e6 < beta

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 63.3%

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

   3. Applied rewrites 63.3%

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

Alternative 2: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq -5200000:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \left(1 - \frac{\alpha}{2 + \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (if (<= beta -5200000.0)
  (- -0.5 -0.5)
  (if (<= beta 2.1e-42)
    (* 0.5 (- 1.0 (/ alpha (+ 2.0 alpha))))
    (fma (/ beta (+ 2.0 beta)) 0.5 0.5))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -5200000.0) {
		tmp = -0.5 - -0.5;
	} else if (beta <= 2.1e-42) {
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -5200000.0)
		tmp = Float64(-0.5 - -0.5);
	elseif (beta <= 2.1e-42)
		tmp = Float64(0.5 * Float64(1.0 - Float64(alpha / Float64(2.0 + alpha))));
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -5200000.0], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[beta, 2.1e-42], N[(0.5 * N[(1.0 - N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < -5.2e6

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if -5.2e6 < beta < 2.1000000000000001e-42

   1. Initial program 63.3%

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

   2. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 58.4%

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

   4. Applied rewrites 58.4%

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

   if 2.1000000000000001e-42 < beta

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 63.3%

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

   3. Applied rewrites 63.3%

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

   4. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1}, 0.5, 0.5\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 43.0%

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

   9. Applied rewrites 43.0%

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

Alternative 3: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -5.4 \cdot 10^{-46}:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+41}:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (if (<= alpha -5.4e-46)
  (- -0.5 -0.5)
  (if (<= alpha 1.6e-16)
    (fma (/ beta (+ 2.0 beta)) 0.5 0.5)
    (if (<= alpha 4.4e+41)
      (- -0.5 -0.5)
      (+ (/ 1.0 alpha) (/ beta alpha))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -5.4e-46) {
		tmp = -0.5 - -0.5;
	} else if (alpha <= 1.6e-16) {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	} else if (alpha <= 4.4e+41) {
		tmp = -0.5 - -0.5;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -5.4e-46)
		tmp = Float64(-0.5 - -0.5);
	elseif (alpha <= 1.6e-16)
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	elseif (alpha <= 4.4e+41)
		tmp = Float64(-0.5 - -0.5);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -5.4e-46], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[alpha, 1.6e-16], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], If[LessEqual[alpha, 4.4e+41], N[(-0.5 - -0.5), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -5.4e-46 or 1.6000000000000001e-16 < alpha < 4.3999999999999998e41

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if -5.4e-46 < alpha < 1.6000000000000001e-16

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 63.3%

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

   3. Applied rewrites 63.3%

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

   4. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1}, 0.5, 0.5\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 43.0%

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

   9. Applied rewrites 43.0%

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

   if 4.3999999999999998e41 < alpha

   1. Initial program 63.3%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

      4. lower-*.f64 17.9%

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   4. Applied rewrites 17.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 17.9%

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

   7. Applied rewrites 17.9%

      \[\leadsto \frac{1}{\alpha} + \color{blue}{\frac{\beta}{\alpha}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.04:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5 + \beta \cdot \left(0.25 + -0.125 \cdot \beta\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 - -0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0
        (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
  (if (<= t_0 0.04)
    (- -0.5 -0.5)
    (if (<= t_0 0.6)
      (+ 0.5 (* beta (+ 0.25 (* -0.125 beta))))
      (- -0.5 -0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.04) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.04d0) then
        tmp = (-0.5d0) - (-0.5d0)
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0 + (beta * (0.25d0 + ((-0.125d0) * beta)))
    else
        tmp = (-0.5d0) - (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.04) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.04:
		tmp = -0.5 - -0.5
	elif t_0 <= 0.6:
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)))
	else:
		tmp = -0.5 - -0.5
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.04)
		tmp = Float64(-0.5 - -0.5);
	elseif (t_0 <= 0.6)
		tmp = Float64(0.5 + Float64(beta * Float64(0.25 + Float64(-0.125 * beta))));
	else
		tmp = Float64(-0.5 - -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.04)
		tmp = -0.5 - -0.5;
	elseif (t_0 <= 0.6)
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	else
		tmp = -0.5 - -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.04], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.5 + N[(beta * N[(0.25 + N[(-0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.040000000000000001 or 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if 0.040000000000000001 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 63.3%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 43.0%

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 26.5%

         \[\leadsto 0.5 + \beta \cdot \left(0.25 + -0.125 \cdot \beta\right) \]

   7. Applied rewrites 26.5%

      \[\leadsto 0.5 + \color{blue}{\beta \cdot \left(0.25 + -0.125 \cdot \beta\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, \alpha, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 - -0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0
        (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
  (if (<= t_0 1e-7)
    (- -0.5 -0.5)
    (if (<= t_0 0.6)
      (fma (fma 0.125 alpha -0.25) alpha 0.5)
      (- -0.5 -0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 0.5);
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = Float64(-0.5 - -0.5);
	elseif (t_0 <= 0.6)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 0.5);
	else
		tmp = Float64(-0.5 - -0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-7], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(-0.5 - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999995e-8 or 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if 9.9999999999999995e-8 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 63.3%

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

   2. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 58.4%

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 28.7%

         \[\leadsto 0.5 + \alpha \cdot \left(0.125 \cdot \alpha - 0.25\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto 0.5 + \color{blue}{\alpha \cdot \left(0.125 \cdot \alpha - 0.25\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 28.7%

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 28.7%

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

   9. Applied rewrites 28.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, \alpha, -0.25\right), \alpha, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;-0.5 - -0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0
        (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
  (if (<= t_0 1e-7)
    (- -0.5 -0.5)
    (if (<= t_0 0.6) (+ 0.5 (* 0.25 beta)) (- -0.5 -0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 1d-7) then
        tmp = (-0.5d0) - (-0.5d0)
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0 + (0.25d0 * beta)
    else
        tmp = (-0.5d0) - (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 1e-7:
		tmp = -0.5 - -0.5
	elif t_0 <= 0.6:
		tmp = 0.5 + (0.25 * beta)
	else:
		tmp = -0.5 - -0.5
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = Float64(-0.5 - -0.5);
	elseif (t_0 <= 0.6)
		tmp = Float64(0.5 + Float64(0.25 * beta));
	else
		tmp = Float64(-0.5 - -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 1e-7)
		tmp = -0.5 - -0.5;
	elseif (t_0 <= 0.6)
		tmp = 0.5 + (0.25 * beta);
	else
		tmp = -0.5 - -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-7], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.5 + N[(0.25 * beta), $MachinePrecision]), $MachinePrecision], N[(-0.5 - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999995e-8 or 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if 9.9999999999999995e-8 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 63.3%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 43.0%

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 27.3%

         \[\leadsto 0.5 + 0.25 \cdot \beta \]

   7. Applied rewrites 27.3%

      \[\leadsto 0.5 + \color{blue}{0.25 \cdot \beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.04:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 - -0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0
        (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
  (if (<= t_0 0.04)
    (- -0.5 -0.5)
    (if (<= t_0 0.6) (fma alpha -0.25 0.5) (- -0.5 -0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.04) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.04)
		tmp = Float64(-0.5 - -0.5);
	elseif (t_0 <= 0.6)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = Float64(-0.5 - -0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.04], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(alpha * -0.25 + 0.5), $MachinePrecision], N[(-0.5 - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.040000000000000001 or 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if 0.040000000000000001 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 63.3%

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

   2. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 58.4%

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 28.4%

         \[\leadsto 0.5 + -0.25 \cdot \alpha \]

   7. Applied rewrites 28.4%

      \[\leadsto 0.5 + \color{blue}{-0.25 \cdot \alpha} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 28.4%

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

   9. Applied rewrites 28.4%

      \[\leadsto \mathsf{fma}\left(\alpha, -0.25, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;-0.5 - -0.5\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 - -0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0
        (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
  (if (<= t_0 1e-7)
    (- -0.5 -0.5)
    (if (<= t_0 0.6) 0.5 (- -0.5 -0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 1d-7) then
        tmp = (-0.5d0) - (-0.5d0)
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = (-0.5d0) - (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = -0.5 - -0.5;
	} else if (t_0 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = -0.5 - -0.5;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 1e-7:
		tmp = -0.5 - -0.5
	elif t_0 <= 0.6:
		tmp = 0.5
	else:
		tmp = -0.5 - -0.5
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = Float64(-0.5 - -0.5);
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = Float64(-0.5 - -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 1e-7)
		tmp = -0.5 - -0.5;
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = -0.5 - -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-7], N[(-0.5 - -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.6], 0.5, N[(-0.5 - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999995e-8 or 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 43.9%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 20.0%

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

   6. Applied rewrites 20.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 20.0%

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{-1}{2}} - -0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

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

   if 9.9999999999999995e-8 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 63.3%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 43.0%

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.6%

      \[\leadsto 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 29.6% accurate, 19.0× speedup

Math

\[0.5 \]

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)
use fmin_fmax_functions
    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 63.3%

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

2. Taylor expanded in alpha around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 43.0%

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

4. Applied rewrites 43.0%

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

5. Taylor expanded in beta around 0

   \[\leadsto \frac{1}{2} \]
6. Step-by-step derivation

7. Applied rewrites 29.6%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(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))