Octave 3.8, jcobi/1

Percentage Accurate: 75.4% → 99.7%

Time: 11.3s

Alternatives: 10

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + 2 \cdot \beta\\ \mathbf{if}\;\beta \leq 440000000000:\\ \;\;\;\;\frac{1}{\alpha \cdot \mathsf{fma}\left(-2, \frac{\left(2 + \beta\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}}{{t\_0}^{2}}, \frac{2}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* 2.0 beta))))
   (if (<= beta 440000000000.0)
     (/
      1.0
      (*
       alpha
       (fma
        -2.0
        (/ (* (+ 2.0 beta) (/ (- (- -2.0 beta) beta) alpha)) (pow t_0 2.0))
        (/ 2.0 t_0))))
     (/ 1.0 (+ 1.0 (/ (+ 1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (2.0 * beta);
	double tmp;
	if (beta <= 440000000000.0) {
		tmp = 1.0 / (alpha * fma(-2.0, (((2.0 + beta) * (((-2.0 - beta) - beta) / alpha)) / pow(t_0, 2.0)), (2.0 / t_0)));
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(2.0 * beta))
	tmp = 0.0
	if (beta <= 440000000000.0)
		tmp = Float64(1.0 / Float64(alpha * fma(-2.0, Float64(Float64(Float64(2.0 + beta) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)) / (t_0 ^ 2.0)), Float64(2.0 / t_0))));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 440000000000.0], N[(1.0 / N[(alpha * N[(-2.0 * N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4e11

   1. Initial program 65.8%

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

      1. +-commutative 65.8%

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

   3. Simplified 65.8%

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

      1. div-inv 65.7%

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

      2. fma-define 65.1%

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

      3. associate-+l+ 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. fma-undefine 65.7%

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

      2. div-inv 65.8%

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

      3. rem-log-exp 65.7%

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

      4. clear-num 65.7%

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

      5. inv-pow 65.7%

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

      6. rem-log-exp 65.8%

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

      7. +-commutative 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. unpow-1 65.8%

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

      2. associate-+r+ 65.8%

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

      3. +-commutative 65.8%

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

      4. +-commutative 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in alpha around inf 99.8%

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

       1. fma-define 99.8%

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

   13. Simplified 99.8%

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

   if 4.4e11 < beta

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. div-inv 88.9%

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

      2. fma-define 88.9%

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

      3. associate-+l+ 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. fma-undefine 88.9%

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

      2. div-inv 88.9%

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

      3. rem-log-exp 88.9%

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

      4. clear-num 88.9%

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

      5. inv-pow 88.9%

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

      6. rem-log-exp 88.9%

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

      7. +-commutative 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. unpow-1 88.9%

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

      2. associate-+r+ 88.9%

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

      3. +-commutative 88.9%

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

      4. +-commutative 88.9%

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

   10. Simplified 88.9%

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

   11. Taylor expanded in alpha around 0 97.1%

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

   12. Taylor expanded in beta around inf 99.9%

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

       1. *-lft-identity 99.9%

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

       2. metadata-eval 99.9%

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

       3. cancel-sign-sub-inv 99.9%

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

       4. associate-*r/ 99.9%

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

       5. div-sub 99.9%

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

       6. sub-neg 99.9%

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

       7. mul-1-neg 99.9%

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

       8. remove-double-neg 99.9%

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

       9. +-commutative 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 440000000000:\\ \;\;\;\;\frac{1}{\alpha \cdot \mathsf{fma}\left(-2, \frac{\left(2 + \beta\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}}{{\left(2 + 2 \cdot \beta\right)}^{2}}, \frac{2}{2 + 2 \cdot \beta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\beta}{2 + \beta}\\ \frac{1}{2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{t\_0}^{2}} + 2 \cdot \frac{1}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ beta (+ 2.0 beta)))))
   (/
    1.0
    (+
     (*
      2.0
      (/
       (* alpha (+ (/ 1.0 (+ 2.0 beta)) (/ beta (pow (+ 2.0 beta) 2.0))))
       (pow t_0 2.0)))
     (* 2.0 (/ 1.0 t_0))))))

C

double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (2.0 + beta));
	return 1.0 / ((2.0 * ((alpha * ((1.0 / (2.0 + beta)) + (beta / pow((2.0 + beta), 2.0)))) / pow(t_0, 2.0))) + (2.0 * (1.0 / t_0)));
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = 1.0d0 + (beta / (2.0d0 + beta))
    code = 1.0d0 / ((2.0d0 * ((alpha * ((1.0d0 / (2.0d0 + beta)) + (beta / ((2.0d0 + beta) ** 2.0d0)))) / (t_0 ** 2.0d0))) + (2.0d0 * (1.0d0 / t_0)))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (2.0 + beta));
	return 1.0 / ((2.0 * ((alpha * ((1.0 / (2.0 + beta)) + (beta / Math.pow((2.0 + beta), 2.0)))) / Math.pow(t_0, 2.0))) + (2.0 * (1.0 / t_0)));
}

Python

def code(alpha, beta):
	t_0 = 1.0 + (beta / (2.0 + beta))
	return 1.0 / ((2.0 * ((alpha * ((1.0 / (2.0 + beta)) + (beta / math.pow((2.0 + beta), 2.0)))) / math.pow(t_0, 2.0))) + (2.0 * (1.0 / t_0)))

Julia

function code(alpha, beta)
	t_0 = Float64(1.0 + Float64(beta / Float64(2.0 + beta)))
	return Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(alpha * Float64(Float64(1.0 / Float64(2.0 + beta)) + Float64(beta / (Float64(2.0 + beta) ^ 2.0)))) / (t_0 ^ 2.0))) + Float64(2.0 * Float64(1.0 / t_0))))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = 1.0 + (beta / (2.0 + beta));
	tmp = 1.0 / ((2.0 * ((alpha * ((1.0 / (2.0 + beta)) + (beta / ((2.0 + beta) ^ 2.0)))) / (t_0 ^ 2.0))) + (2.0 * (1.0 / t_0)));
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. +-commutative 73.8%

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

3. Simplified 73.8%

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

   1. div-inv 73.8%

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

   2. fma-define 73.4%

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

   3. associate-+l+ 73.4%

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

6. Applied egg-rr 73.4%

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

   1. fma-undefine 73.8%

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

   2. div-inv 73.8%

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

   3. rem-log-exp 73.8%

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

   4. clear-num 73.8%

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

   5. inv-pow 73.8%

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

   6. rem-log-exp 73.8%

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

   7. +-commutative 73.8%

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

8. Applied egg-rr 73.8%

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

   1. unpow-1 73.8%

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

   2. associate-+r+ 73.8%

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

   3. +-commutative 73.8%

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

   4. +-commutative 73.8%

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

10. Simplified 73.8%

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

11. Taylor expanded in alpha around 0 99.0%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 5.4%

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

      1. +-commutative 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in alpha around inf 100.0%

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

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

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-+l- 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

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

Alternative 4: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 5.4%

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

      1. +-commutative 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in alpha around inf 100.0%

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

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

   1. Initial program 99.6%

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

4. Final simplification 99.7%

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

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

   3. Simplified 66.5%

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

      1. div-inv 66.4%

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

      2. fma-define 65.9%

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

      3. associate-+l+ 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. fma-undefine 66.4%

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

      2. div-inv 66.5%

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

      3. rem-log-exp 66.5%

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

      4. clear-num 66.5%

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

      5. inv-pow 66.5%

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

      6. rem-log-exp 66.5%

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

      7. +-commutative 66.5%

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

   8. Applied egg-rr 66.5%

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

      1. unpow-1 66.5%

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

      2. associate-+r+ 66.5%

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

      3. +-commutative 66.5%

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

      4. +-commutative 66.5%

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

   10. Simplified 66.5%

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

   11. Taylor expanded in alpha around 0 100.0%

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

   12. Taylor expanded in beta around 0 98.5%

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

       1. +-commutative 98.5%

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

   14. Simplified 98.5%

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

   if 1 < beta

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. div-inv 87.0%

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

      2. fma-define 86.9%

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

      3. associate-+l+ 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. fma-undefine 87.0%

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

      2. div-inv 87.0%

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

      3. rem-log-exp 87.0%

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

      4. clear-num 87.0%

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

      5. inv-pow 87.0%

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

      6. rem-log-exp 87.0%

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

      7. +-commutative 87.0%

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

   8. Applied egg-rr 87.0%

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

      1. unpow-1 87.0%

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

      2. associate-+r+ 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

   10. Simplified 87.0%

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

   11. Taylor expanded in alpha around 0 97.2%

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

   12. Taylor expanded in beta around inf 98.8%

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

       1. *-lft-identity 98.8%

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

       2. metadata-eval 98.8%

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

       3. cancel-sign-sub-inv 98.8%

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

       4. associate-*r/ 98.8%

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

       5. div-sub 98.8%

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

       6. sub-neg 98.8%

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

       7. mul-1-neg 98.8%

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

       8. remove-double-neg 98.8%

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

       9. +-commutative 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 98.6%

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

Alternative 6: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3850000:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3850000.0) (/ 1.0 (+ 2.0 alpha)) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3850000.0) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3850000.0d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3850000.0) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3850000.0:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3850000.0)
		tmp = Float64(1.0 / Float64(2.0 + alpha));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3850000.0)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3850000.0], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.85e6

   1. Initial program 65.8%

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

      1. +-commutative 65.8%

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

   3. Simplified 65.8%

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

      1. div-inv 65.7%

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

      2. fma-define 65.1%

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

      3. associate-+l+ 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. fma-undefine 65.7%

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

      2. div-inv 65.8%

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

      3. rem-log-exp 65.7%

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

      4. clear-num 65.7%

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

      5. inv-pow 65.7%

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

      6. rem-log-exp 65.8%

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

      7. +-commutative 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. unpow-1 65.8%

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

      2. associate-+r+ 65.8%

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

      3. +-commutative 65.8%

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

      4. +-commutative 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in alpha around 0 100.0%

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

   12. Taylor expanded in beta around 0 97.5%

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

       1. +-commutative 97.5%

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

   14. Simplified 97.5%

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

   if 3.85e6 < beta

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in alpha around 0 88.4%

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

   6. Taylor expanded in beta around inf 88.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3850000:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in alpha around 0 62.3%

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

   6. Taylor expanded in beta around 0 61.5%

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

      1. *-commutative 61.5%

         \[\leadsto 0.5 + \color{blue}{\beta \cdot 0.25} \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{0.5 + \beta \cdot 0.25} \]

   if 2 < beta

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in alpha around 0 86.6%

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

   6. Taylor expanded in beta around inf 86.6%

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

Alternative 8: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in alpha around 0 62.3%

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

   6. Taylor expanded in beta around 0 61.5%

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

      1. *-commutative 61.5%

         \[\leadsto 0.5 + \color{blue}{\beta \cdot 0.25} \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{0.5 + \beta \cdot 0.25} \]

   if 2 < beta

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. div-inv 87.0%

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

      2. fma-define 86.9%

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

      3. associate-+l+ 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. fma-undefine 87.0%

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

      2. div-inv 87.0%

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

      3. rem-log-exp 87.0%

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

      4. clear-num 87.0%

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

      5. inv-pow 87.0%

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

      6. rem-log-exp 87.0%

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

      7. +-commutative 87.0%

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

   8. Applied egg-rr 87.0%

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

      1. unpow-1 87.0%

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

      2. associate-+r+ 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

   10. Simplified 87.0%

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

   11. Taylor expanded in alpha around 0 97.2%

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

   12. Taylor expanded in beta around inf 86.4%

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

Alternative 9: 72.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3850000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 3850000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3850000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3850000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3850000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3850000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3850000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3850000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3850000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 3.85e6

   1. Initial program 65.8%

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

      1. +-commutative 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in beta around 0 64.3%

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

      1. +-commutative 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in alpha around 0 60.1%

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

   if 3.85e6 < beta

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. div-inv 88.9%

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

      2. fma-define 88.9%

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

      3. associate-+l+ 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. fma-undefine 88.9%

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

      2. div-inv 88.9%

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

      3. rem-log-exp 88.9%

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

      4. clear-num 88.9%

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

      5. inv-pow 88.9%

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

      6. rem-log-exp 88.9%

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

      7. +-commutative 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. unpow-1 88.9%

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

      2. associate-+r+ 88.9%

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

      3. +-commutative 88.9%

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

      4. +-commutative 88.9%

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

   10. Simplified 88.9%

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

   11. Taylor expanded in alpha around 0 97.1%

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

   12. Taylor expanded in beta around inf 88.2%

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

Alternative 10: 49.8% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 73.8%

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

   1. +-commutative 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in beta around 0 47.0%

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

   1. +-commutative 47.0%

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

7. Simplified 47.0%

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

8. Taylor expanded in alpha around 0 45.2%

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

Reproduce

herbie shell --seed 2024125 
(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))