Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 75.5%

Time: 13.2s

Alternatives: 5

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: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 75.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. div-sub N/A

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

   2. associate-+l- N/A

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

   3. --lowering--.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-lowering-+.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. +-commutative N/A

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

   14. associate-+l+ N/A

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

   15. +-lowering-+.f64 N/A

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

   16. +-lowering-+.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. +-lowering-+.f64 77.1

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

6. Applied egg-rr 77.1%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-lowering-+.f64 77.1

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

8. Applied egg-rr 77.1%

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

Alternative 2: 75.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. div-sub N/A

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

   2. associate-+l- N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. neg-lowering-neg.f64 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-lowering-+.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. +-commutative N/A

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

   17. associate-+l+ N/A

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

   18. +-lowering-+.f64 N/A

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

   19. +-lowering-+.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta, \color{blue}{0 - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}\right)}{2} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta, 0 - \color{blue}{\left(-1 + \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}\right)}{2} \]

   8. associate--r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta, \color{blue}{\left(0 - -1\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}\right)}{2} \]

   9. metadata-eval N/A

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

   10. --lowering--.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. +-lowering-+.f64 77.1

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

6. Applied egg-rr 77.1%

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

Alternative 3: 75.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return fma(Float64(beta / t_0), 0.5, Float64(0.5 * Float64(Float64(Float64(-alpha) / t_0) - -1.0)))
end

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. div-sub N/A

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

   2. associate-+l- N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. neg-lowering-neg.f64 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-lowering-+.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. +-commutative N/A

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

   17. associate-+l+ N/A

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

   18. +-lowering-+.f64 N/A

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

   19. +-lowering-+.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. div-inv N/A

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

   2. sub-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. neg-lowering-neg.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. *-lowering-*.f64 N/A

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

6. Applied egg-rr 77.1%

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

7. Final simplification 77.1%

   \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5 \cdot \left(\frac{-\alpha}{\beta + \left(\alpha + 2\right)} - -1\right)\right) \]
8. Add Preprocessing

Alternative 4: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right) \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (fma (/ (- beta alpha) (+ beta (+ alpha 2.0))) 0.5 0.5))

C

double code(double alpha, double beta) {
	return fma(((beta - alpha) / (beta + (alpha + 2.0))), 0.5, 0.5);
}

Julia

function code(alpha, beta)
	return fma(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))), 0.5, 0.5)
end

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

   12. +-lowering-+.f64 N/A

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

   13. +-lowering-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 76.4

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

4. Applied egg-rr 76.4%

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

Alternative 5: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right) \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (fma (- beta alpha) (/ 0.5 (+ beta (+ alpha 2.0))) 0.5))

C

double code(double alpha, double beta) {
	return fma((beta - alpha), (0.5 / (beta + (alpha + 2.0))), 0.5);
}

Julia

function code(alpha, beta)
	return fma(Float64(beta - alpha), Float64(0.5 / Float64(beta + Float64(alpha + 2.0))), 0.5)
end

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. div-sub N/A

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

   2. associate-+l- N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. neg-lowering-neg.f64 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-lowering-+.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. +-commutative N/A

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

   17. associate-+l+ N/A

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

   18. +-lowering-+.f64 N/A

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

   19. +-lowering-+.f64 77.1

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

4. Applied egg-rr 77.1%

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

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft1-in N/A

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

   10. associate-*l/ N/A

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

   11. associate-/l* N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 76.3%

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

Reproduce

herbie shell --seed 2024208 
(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))