Octave 3.8, jcobi/2

Percentage Accurate: 62.1% → 97.7%

Time: 9.8s

Alternatives: 13

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: 62.1% 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: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, {\left(\frac{t\_1}{\frac{\alpha + \beta}{t\_1 + 2}}\right)}^{-1}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (fma 2.0 i (+ alpha beta))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) -0.99999)
     (* (/ (- (- -2.0 (fma 2.0 i beta)) (fma 2.0 i beta)) alpha) -0.5)
     (/
      (fma
       (- beta alpha)
       (pow (/ t_1 (/ (+ alpha beta) (+ t_1 2.0))) -1.0)
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.99999) {
		tmp = (((-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5;
	} else {
		tmp = fma((beta - alpha), pow((t_1 / ((alpha + beta) / (t_1 + 2.0))), -1.0), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5);
	else
		tmp = Float64(fma(Float64(beta - alpha), (Float64(t_1 / Float64(Float64(alpha + beta) / Float64(t_1 + 2.0))) ^ -1.0), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[Power[N[(t$95$1 / N[(N[(alpha + beta), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $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))) < -0.999990000000000046

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -0.999990000000000046 < (/.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 75.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}\right)}^{-1}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (* (/ (- (- -2.0 (fma 2.0 i beta)) (fma 2.0 i beta)) alpha) -0.5)
     (if (<= t_1 2e-20) 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 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (((-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5;
	} else if (t_1 <= 2e-20) {
		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(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(Float64(Float64(-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5);
	elseif (t_1 <= 2e-20)
		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[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $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[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.5

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   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))) < 1.99999999999999989e-20

   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 99.4%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in 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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (* (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha) 0.5)
     (if (<= t_1 2e-20) 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 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-20) {
		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(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-20)
		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[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $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[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.5

   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. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   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))) < 1.99999999999999989e-20

   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 99.4%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in 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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, i, -\mathsf{fma}\left(2, i, 2\right)\right)}{\alpha} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9)
     (* (/ (fma -2.0 i (- (fma 2.0 i 2.0))) alpha) -0.5)
     (if (<= t_1 2e-20) 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 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9) {
		tmp = (fma(-2.0, i, -fma(2.0, i, 2.0)) / alpha) * -0.5;
	} else if (t_1 <= 2e-20) {
		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(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(Float64(fma(-2.0, i, Float64(-fma(2.0, i, 2.0))) / alpha) * -0.5);
	elseif (t_1 <= 2e-20)
		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[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(N[(N[(-2.0 * i + (-N[(2.0 * i + 2.0), $MachinePrecision])), $MachinePrecision] / alpha), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.900000000000000022

   1. Initial program 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 17.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 17.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 17.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in alpha around -inf

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

   10. Applied rewrites 77.9%

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

   if -0.900000000000000022 < (/.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))) < 1.99999999999999989e-20

   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 98.7%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in 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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, i, -\mathsf{fma}\left(2, i, 2\right)\right)}{\alpha} \cdot -0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 2e-20) 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 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-20) {
		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(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-20)
		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[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.900000000000000022

   1. Initial program 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 17.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 17.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 17.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in alpha around inf

      \[\leadsto \frac{2 + 4 \cdot i}{\alpha} \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.9%

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

   if -0.900000000000000022 < (/.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))) < 1.99999999999999989e-20

   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 98.7%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in 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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, i, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4 \cdot i}{\beta}, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 2e-20) 0.5 (fma (/ (* -4.0 i) beta) 0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-20) {
		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(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-20)
		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[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.900000000000000022

   1. Initial program 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 17.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 17.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 17.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in alpha around inf

      \[\leadsto \frac{2 + 4 \cdot i}{\alpha} \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.9%

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

   if -0.900000000000000022 < (/.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))) < 1.99999999999999989e-20

   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 98.7%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in 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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 88.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4 \cdot i}{\beta}, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta}{2 + \beta}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 1e-24) 0.5 (* (+ 1.0 (/ beta (+ 2.0 beta))) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 1e-24) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + (beta / (2.0 + beta))) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 1e-24)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + beta))) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e-24], 0.5, N[(N[(1.0 + N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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))) < -0.900000000000000022

   1. Initial program 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 17.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 17.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 17.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in alpha around inf

      \[\leadsto \frac{2 + 4 \cdot i}{\alpha} \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.9%

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

   if -0.900000000000000022 < (/.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))) < 9.99999999999999924e-25

   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 98.7%

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

   if 9.99999999999999924e-25 < (/.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 30.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.0%

      \[\leadsto \left(1 + \frac{\beta}{2 + \beta}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta}{2 + \beta}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.9)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 2e-20) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.9) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-20) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-20)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], 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))) < -0.900000000000000022

   1. Initial program 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 17.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 17.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 17.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in alpha around inf

      \[\leadsto \frac{2 + 4 \cdot i}{\alpha} \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.9%

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

   if -0.900000000000000022 < (/.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))) < 1.99999999999999989e-20

   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 98.7%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 88.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta + \alpha}{t\_1}, \frac{\beta - \alpha}{t\_1 + 2}, 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (fma i 2.0 (+ beta alpha))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) -0.99999)
     (* (/ (- (- -2.0 (fma 2.0 i beta)) (fma 2.0 i beta)) alpha) -0.5)
     (* (fma (/ (+ beta alpha) t_1) (/ (- beta alpha) (+ t_1 2.0)) 1.0) 0.5))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.99999) {
		tmp = (((-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5;
	} else {
		tmp = fma(((beta + alpha) / t_1), ((beta - alpha) / (t_1 + 2.0)), 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5);
	else
		tmp = Float64(fma(Float64(Float64(beta + alpha) / t_1), Float64(Float64(beta - alpha) / Float64(t_1 + 2.0)), 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(beta + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(t$95$1 + 2.0), $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))) < -0.999990000000000046

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -0.999990000000000046 < (/.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 75.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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. div-inv N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2}, 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   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 75.5%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 97.2%

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

Alternative 11: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 + 2\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ t_0 2.0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.99999)
     (* (/ (- (- -2.0 (fma 2.0 i beta)) (fma 2.0 i beta)) alpha) -0.5)
     (/ (+ (/ (- beta alpha) t_1) 1.0) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 + 2.0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999) {
		tmp = (((-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5;
	} else {
		tmp = (((beta - alpha) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(-2.0 - fma(2.0, i, beta)) - fma(2.0, i, beta)) / alpha) * -0.5);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / t_1) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.99999], N[(N[(N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $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))) < -0.999990000000000046

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -0.999990000000000046 < (/.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 75.7%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 2e-20)
     0.5
     1.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= 2e-20) {
		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 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) <= 2d-20) 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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= 2e-20) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= 2e-20:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) <= 2e-20)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= 2e-20)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], 2e-20], 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))) < 1.99999999999999989e-20

   1. Initial program 71.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 72.4%

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

   if 1.99999999999999989e-20 < (/.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 29.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 88.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.2% 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 59.9%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 59.9%

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

6. Final simplification 59.9%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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))