Octave 3.8, jcobi/2

Percentage Accurate: 62.4% → 98.0%

Time: 10.8s

Alternatives: 19

Speedup: 1.1×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2} \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\beta + \alpha}{t\_1} \cdot \left(\beta - \alpha\right)}{t\_1 + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha))) (t_1 (fma i 2.0 (+ beta alpha))))
   (if (<= (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0)) -0.99995)
     (fma (/ (fma beta 2.0 2.0) alpha) 0.5 (* (/ i alpha) 2.0))
     (fma (/ (* (/ (+ beta alpha) t_1) (- beta alpha)) (+ t_1 2.0)) 0.5 0.5))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (((((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0)) <= -0.99995) {
		tmp = fma((fma(beta, 2.0, 2.0) / alpha), 0.5, ((i / alpha) * 2.0));
	} else {
		tmp = fma(((((beta + alpha) / t_1) * (beta - alpha)) / (t_1 + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0)) <= -0.99995)
		tmp = fma(Float64(fma(beta, 2.0, 2.0) / alpha), 0.5, Float64(Float64(i / alpha) * 2.0));
	else
		tmp = fma(Float64(Float64(Float64(Float64(beta + alpha) / t_1) * Float64(beta - alpha)) / Float64(t_1 + 2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5 + N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.0%

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

   if -0.999950000000000006 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_2 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_1}}{t\_1 + 2}\\ \mathbf{if}\;t\_2 \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;t\_2 \leq 0.99999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{\beta + \alpha}{\left(t\_0 + 2\right) \cdot t\_0}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right) + 2} \cdot \frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha)))
        (t_1 (+ (* i 2.0) (+ beta alpha)))
        (t_2 (/ (/ (* (- beta alpha) (+ beta alpha)) t_1) (+ t_1 2.0))))
   (if (<= t_2 -0.99995)
     (fma (/ (fma beta 2.0 2.0) alpha) 0.5 (* (/ i alpha) 2.0))
     (if (<= t_2 0.99999999)
       (/ (fma (- beta alpha) (/ (+ beta alpha) (* (+ t_0 2.0) t_0)) 1.0) 2.0)
       (fma
        (* (/ beta (+ (fma i 2.0 beta) 2.0)) (/ beta (fma i 2.0 beta)))
        0.5
        0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = (i * 2.0) + (beta + alpha);
	double t_2 = (((beta - alpha) * (beta + alpha)) / t_1) / (t_1 + 2.0);
	double tmp;
	if (t_2 <= -0.99995) {
		tmp = fma((fma(beta, 2.0, 2.0) / alpha), 0.5, ((i / alpha) * 2.0));
	} else if (t_2 <= 0.99999999) {
		tmp = fma((beta - alpha), ((beta + alpha) / ((t_0 + 2.0) * t_0)), 1.0) / 2.0;
	} else {
		tmp = fma(((beta / (fma(i, 2.0, beta) + 2.0)) * (beta / fma(i, 2.0, beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_2 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_1) / Float64(t_1 + 2.0))
	tmp = 0.0
	if (t_2 <= -0.99995)
		tmp = fma(Float64(fma(beta, 2.0, 2.0) / alpha), 0.5, Float64(Float64(i / alpha) * 2.0));
	elseif (t_2 <= 0.99999999)
		tmp = Float64(fma(Float64(beta - alpha), Float64(Float64(beta + alpha) / Float64(Float64(t_0 + 2.0) * t_0)), 1.0) / 2.0);
	else
		tmp = fma(Float64(Float64(beta / Float64(fma(i, 2.0, beta) + 2.0)) * Float64(beta / fma(i, 2.0, beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99995], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5 + N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99999999], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(N[(t$95$0 + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(N[(i * 2.0 + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.0%

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

   if -0.999950000000000006 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.99999998999999995

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 0.99999998999999995 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 28.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.99999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{\beta + \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right) + 2} \cdot \frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\frac{\alpha}{\mathsf{fma}\left(2, i, \alpha\right)} \cdot \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + 2}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99995)
     (fma (/ (fma beta 2.0 2.0) alpha) 0.5 (* (/ i alpha) 2.0))
     (if (<= t_1 0.002)
       (fma
        -0.5
        (/ (* (/ alpha (fma 2.0 i alpha)) alpha) (+ (fma 2.0 i alpha) 2.0))
        0.5)
       (fma (/ (- -2.0 (fma 4.0 i (* 2.0 alpha))) beta) 0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99995) {
		tmp = fma((fma(beta, 2.0, 2.0) / alpha), 0.5, ((i / alpha) * 2.0));
	} else if (t_1 <= 0.002) {
		tmp = fma(-0.5, (((alpha / fma(2.0, i, alpha)) * alpha) / (fma(2.0, i, alpha) + 2.0)), 0.5);
	} else {
		tmp = fma(((-2.0 - fma(4.0, i, (2.0 * alpha))) / beta), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99995)
		tmp = fma(Float64(fma(beta, 2.0, 2.0) / alpha), 0.5, Float64(Float64(i / alpha) * 2.0));
	elseif (t_1 <= 0.002)
		tmp = fma(-0.5, Float64(Float64(Float64(alpha / fma(2.0, i, alpha)) * alpha) / Float64(fma(2.0, i, alpha) + 2.0)), 0.5);
	else
		tmp = fma(Float64(Float64(-2.0 - fma(4.0, i, Float64(2.0 * alpha))) / beta), 0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99995], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5 + N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(-0.5 * N[(N[(N[(alpha / N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] * alpha), $MachinePrecision] / N[(N[(2.0 * i + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-2.0 - N[(4.0 * i + N[(2.0 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.0%

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

   if -0.999950000000000006 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.2

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

   9. Applied rewrites 99.2%

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

   11. Applied rewrites 99.2%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--r+ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} \cdot \alpha - 2\right) - \left(2 \cdot \alpha + 4 \cdot i\right)}{\beta}, \frac{1}{2}, 1\right) \]

      8. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - 2\right) - \left(2 \cdot \alpha + 4 \cdot i\right)}{\beta}, \frac{1}{2}, 1\right) \]

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.4

          \[\leadsto \mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, \color{blue}{2 \cdot \alpha}\right)}{\beta}, 0.5, 1\right) \]

   5. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\frac{\alpha}{\mathsf{fma}\left(2, i, \alpha\right)} \cdot \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + 2}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99995)
     (fma (/ (fma beta 2.0 2.0) alpha) 0.5 (* (/ i alpha) 2.0))
     (if (<= t_1 0.002)
       (fma
        0.5
        (/ (* alpha alpha) (* (- -2.0 (fma i 2.0 alpha)) (fma i 2.0 alpha)))
        0.5)
       (fma (/ (- -2.0 (fma 4.0 i (* 2.0 alpha))) beta) 0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99995) {
		tmp = fma((fma(beta, 2.0, 2.0) / alpha), 0.5, ((i / alpha) * 2.0));
	} else if (t_1 <= 0.002) {
		tmp = fma(0.5, ((alpha * alpha) / ((-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 0.5);
	} else {
		tmp = fma(((-2.0 - fma(4.0, i, (2.0 * alpha))) / beta), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99995)
		tmp = fma(Float64(fma(beta, 2.0, 2.0) / alpha), 0.5, Float64(Float64(i / alpha) * 2.0));
	elseif (t_1 <= 0.002)
		tmp = fma(0.5, Float64(Float64(alpha * alpha) / Float64(Float64(-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 0.5);
	else
		tmp = fma(Float64(Float64(-2.0 - fma(4.0, i, Float64(2.0 * alpha))) / beta), 0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99995], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5 + N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(0.5 * N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(-2.0 - N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-2.0 - N[(4.0 * i + N[(2.0 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.0%

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

   if -0.999950000000000006 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.7%

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

   3. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--r+ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} \cdot \alpha - 2\right) - \left(2 \cdot \alpha + 4 \cdot i\right)}{\beta}, \frac{1}{2}, 1\right) \]

      8. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{0} - 2\right) - \left(2 \cdot \alpha + 4 \cdot i\right)}{\beta}, \frac{1}{2}, 1\right) \]

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.4

          \[\leadsto \mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, \color{blue}{2 \cdot \alpha}\right)}{\beta}, 0.5, 1\right) \]

   5. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99995:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 - \mathsf{fma}\left(4, i, 2 \cdot \alpha\right)}{\beta}, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (fma (/ (fma beta 2.0 2.0) alpha) 0.5 (* (/ i alpha) 2.0))
     (if (<= t_1 5e-84)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma((fma(beta, 2.0, 2.0) / alpha), 0.5, ((i / alpha) * 2.0));
	} else if (t_1 <= 5e-84) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = fma(Float64(fma(beta, 2.0, 2.0) / alpha), 0.5, Float64(Float64(i / alpha) * 2.0));
	elseif (t_1 <= 5e-84)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5 + N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-84], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 92.0%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000002e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.6%

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

   if 5.0000000000000002e-84 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha}, 0.5, \frac{i}{\alpha} \cdot 2\right)\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right), 0.5, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (/ (fma (fma 4.0 i (* 2.0 beta)) 0.5 1.0) alpha)
     (if (<= t_1 5e-84)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma(fma(4.0, i, (2.0 * beta)), 0.5, 1.0) / alpha;
	} else if (t_1 <= 5e-84) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(fma(fma(4.0, i, Float64(2.0 * beta)), 0.5, 1.0) / alpha);
	elseif (t_1 <= 5e-84)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-84], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(4, i, \beta \cdot 2\right), 0.5, 1\right)}{\color{blue}{\alpha}} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000002e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.6%

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

   if 5.0000000000000002e-84 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right), 0.5, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{0.5}{\alpha} \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(\beta, 2, 2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (* (/ 0.5 alpha) (fma 4.0 i (fma beta 2.0 2.0)))
     (if (<= t_1 5e-84)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (0.5 / alpha) * fma(4.0, i, fma(beta, 2.0, 2.0));
	} else if (t_1 <= 5e-84) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(0.5 / alpha) * fma(4.0, i, fma(beta, 2.0, 2.0)));
	elseif (t_1 <= 5e-84)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(0.5 / alpha), $MachinePrecision] * N[(4.0 * i + N[(beta * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-84], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(4, i, \mathsf{fma}\left(\beta, 2, 2\right)\right) \cdot \color{blue}{\frac{0.5}{\alpha}} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000002e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.6%

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

   if 5.0000000000000002e-84 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{0.5}{\alpha} \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(\beta, 2, 2\right)\right)\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99992)
     (* (/ (fma beta 2.0 2.0) alpha) 0.5)
     (if (<= t_1 5e-84)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99992) {
		tmp = (fma(beta, 2.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 5e-84) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99992)
		tmp = Float64(Float64(fma(beta, 2.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 5e-84)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99992], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-84], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999920000000000031

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 78.0%

      \[\leadsto \frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5 \]

   if -0.999920000000000031 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000002e-84

   1. Initial program 99.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 96.9%

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

   if 5.0000000000000002e-84 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99992)
     (* (/ (fma beta 2.0 2.0) alpha) 0.5)
     (if (<= t_1 0.002) 0.5 (fma (/ (fma 4.0 i 2.0) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99992) {
		tmp = (fma(beta, 2.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = fma((fma(4.0, i, 2.0) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99992)
		tmp = Float64(Float64(fma(beta, 2.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = fma(Float64(fma(4.0, i, 2.0) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99992], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999920000000000031

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 78.0%

      \[\leadsto \frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5 \]

   if -0.999920000000000031 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 96.3%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, \color{blue}{-0.5}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99992)
     (* (/ (fma beta 2.0 2.0) alpha) 0.5)
     (if (<= t_1 0.002) 0.5 (fma (/ (* 4.0 i) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99992) {
		tmp = (fma(beta, 2.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = fma(((4.0 * i) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99992)
		tmp = Float64(Float64(fma(beta, 2.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(4.0 * i) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99992], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, N[(N[(N[(4.0 * i), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999920000000000031

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 78.0%

      \[\leadsto \frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5 \]

   if -0.999920000000000031 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 96.3%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

       \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99992:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.99992:\\ \;\;\;\;\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{0.5}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.99992)
     (* (fma 2.0 beta 2.0) (/ 0.5 alpha))
     (if (<= t_1 0.002) 0.5 (fma (/ (* 4.0 i) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.99992) {
		tmp = fma(2.0, beta, 2.0) * (0.5 / alpha);
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = fma(((4.0 * i) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.99992)
		tmp = Float64(fma(2.0, beta, 2.0) * Float64(0.5 / alpha));
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(4.0 * i) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99992], N[(N[(2.0 * beta + 2.0), $MachinePrecision] * N[(0.5 / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, N[(N[(N[(4.0 * i), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999920000000000031

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 78.0%

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

   10. Applied rewrites 78.0%

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

   if -0.999920000000000031 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 96.3%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

       \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.99992:\\ \;\;\;\;\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{0.5}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9999999999999)
     (/ beta alpha)
     (if (<= t_1 0.002) 0.5 (fma (/ (* 4.0 i) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = fma(((4.0 * i) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9999999999999)
		tmp = Float64(beta / alpha);
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(4.0 * i) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999999999999], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, N[(N[(N[(4.0 * i), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999999999899969

   1. Initial program 2.8%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.1

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 28.0%

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

   if -0.999999999999899969 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 95.0%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

       \[\leadsto \mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4 \cdot i}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9999999999999)
     (/ beta alpha)
     (if (<= t_1 5e-84) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 5e-84) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9999999999999)
		tmp = Float64(beta / alpha);
	elseif (t_1 <= 5e-84)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999999999999], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-84], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999999999899969

   1. Initial program 2.8%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.1

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 28.0%

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

   if -0.999999999999899969 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000002e-84

   1. Initial program 99.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 95.4%

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

   if 5.0000000000000002e-84 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.1

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

      5. lift-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 97.8

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in i around 0

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

   12. Applied rewrites 92.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9999999999999)
     (/ beta alpha)
     (if (<= t_1 0.002) 0.5 (- 1.0 (/ 1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (i * 2.0d0) + (beta + alpha)
    t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0d0)
    if (t_1 <= (-0.9999999999999d0)) then
        tmp = beta / alpha
    else if (t_1 <= 0.002d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (i * 2.0) + (beta + alpha)
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0)
	tmp = 0
	if t_1 <= -0.9999999999999:
		tmp = beta / alpha
	elif t_1 <= 0.002:
		tmp = 0.5
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9999999999999)
		tmp = Float64(beta / alpha);
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (beta + alpha);
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.9999999999999)
		tmp = beta / alpha;
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999999999999], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999999999899969

   1. Initial program 2.8%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.1

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 28.0%

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

   if -0.999999999999899969 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 95.0%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 92.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9999999999999) (/ beta alpha) (if (<= t_1 0.002) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (i * 2.0d0) + (beta + alpha)
    t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0d0)
    if (t_1 <= (-0.9999999999999d0)) then
        tmp = beta / alpha
    else if (t_1 <= 0.002d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9999999999999) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.002) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (i * 2.0) + (beta + alpha)
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0)
	tmp = 0
	if t_1 <= -0.9999999999999:
		tmp = beta / alpha
	elif t_1 <= 0.002:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9999999999999)
		tmp = Float64(beta / alpha);
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (beta + alpha);
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.9999999999999)
		tmp = beta / alpha;
	elseif (t_1 <= 0.002)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999999999999], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999999999899969

   1. Initial program 2.8%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.1

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 28.0%

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

   if -0.999999999999899969 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2e-3

   1. Initial program 99.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 95.0%

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

   if 2e-3 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.9999999999999:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 92.0%

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

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

   1. Initial program 75.1%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 96.7%

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

Alternative 17: 97.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha} \cdot \frac{1}{2} \]

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}\right)}{\alpha} \cdot \frac{1}{2} \]

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.0

          \[\leadsto \frac{\mathsf{fma}\left(4, i, \color{blue}{\beta \cdot 2}\right) + 2}{\alpha} \cdot 0.5 \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, i, \beta \cdot 2\right) + 2}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 92.0%

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

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

   1. Initial program 75.1%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 96.7%

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

Alternative 18: 76.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.5%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 71.7%

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

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

   1. Initial program 30.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.9%

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

4. Final simplification 77.7%

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

Alternative 19: 60.9% accurate, 73.0× speedup

Math

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

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 57.8%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))