Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 95.4%

Time: 8.5s

Alternatives: 10

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: 62.7% 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: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, \alpha, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

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 5e-6)
       (/ (+ t_1 1.0) 2.0)
       (fma (/ (fma 2.0 alpha 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 <= 5e-6) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = fma((fma(2.0, alpha, 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 <= 5e-6)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = fma(Float64(fma(2.0, alpha, 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, 5e-6], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * alpha + 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 4.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   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))) < 5.00000000000000041e-6

   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

   if 5.00000000000000041e-6 < (/.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 29.1%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 86.9%

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

Alternative 2: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(2 + \beta\right) + \alpha}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 1e-41)
       0.5
       (fma (/ (- beta alpha) (+ (+ 2.0 beta) alpha)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (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 <= 1e-41) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / ((2.0 + beta) + alpha)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(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 <= 1e-41)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(2.0 + beta) + alpha)), 0.5, 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, -0.5], 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, 1e-41], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   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))) < 1.00000000000000001e-41

   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 1.00000000000000001e-41 < (/.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 35.9%

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

   3. Taylor expanded in i around 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 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

Alternative 3: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right) \cdot \frac{0.5}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(2 + \beta\right) + \alpha}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 (fma 2.0 beta 2.0)) (/ 0.5 alpha))
     (if (<= t_1 1e-41)
       0.5
       (fma (/ (- beta alpha) (+ (+ 2.0 beta) alpha)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (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, fma(2.0, beta, 2.0)) * (0.5 / alpha);
	} else if (t_1 <= 1e-41) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / ((2.0 + beta) + alpha)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(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(fma(4.0, i, fma(2.0, beta, 2.0)) * Float64(0.5 / alpha));
	elseif (t_1 <= 1e-41)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(2.0 + beta) + alpha)), 0.5, 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, -0.5], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-41], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 91.3%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000001e-41

   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 1.00000000000000001e-41 < (/.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 35.9%

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

   3. Taylor expanded in i around 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 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

Alternative 4: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(2 + \beta\right) + \alpha}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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) alpha) 0.5)
     (if (<= t_1 1e-41)
       0.5
       (fma (/ (- beta alpha) (+ (+ 2.0 beta) alpha)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (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) / alpha) * 0.5;
	} else if (t_1 <= 1e-41) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / ((2.0 + beta) + alpha)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(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(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 1e-41)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(2.0 + beta) + alpha)), 0.5, 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, -0.5], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e-41], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 79.6%

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

   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))) < 1.00000000000000001e-41

   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 1.00000000000000001e-41 < (/.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 35.9%

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

   3. Taylor expanded in i around 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 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

Alternative 5: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ 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) alpha) 0.5)
     (if (<= t_1 1e-41) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 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 <= -0.5) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 1e-41) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(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(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 1e-41)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(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[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e-41], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 79.6%

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

   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))) < 1.00000000000000001e-41

   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 1.00000000000000001e-41 < (/.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 35.9%

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

   3. Taylor expanded in i around 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 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 88.0%

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

Alternative 6: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999999998:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999999999998)
     (* (/ i alpha) 2.0)
     (if (<= t_1 1e-41) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 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 <= -0.999999999998) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 1e-41) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(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 <= -0.999999999998)
		tmp = Float64(Float64(i / alpha) * 2.0);
	elseif (t_1 <= 1e-41)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(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, -0.999999999998], N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-41], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999999999800004

   1. Initial program 3.2%

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

   3. Taylor expanded in 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 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 25.5%

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

   if -0.99999999999800004 < (/.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))) < 1.00000000000000001e-41

   1. Initial program 99.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.8%

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

   if 1.00000000000000001e-41 < (/.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 35.9%

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

   3. Taylor expanded in i around 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 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 88.0%

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

Alternative 7: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999999998:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -0.999999999998)
     (* (/ i alpha) 2.0)
     (if (<= t_1 5e-6) 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 <= -0.999999999998) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 5e-6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)
    if (t_1 <= (-0.999999999998d0)) then
        tmp = (i / alpha) * 2.0d0
    else if (t_1 <= 5d-6) 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 t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999999999998) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 5e-6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)
	tmp = 0
	if t_1 <= -0.999999999998:
		tmp = (i / alpha) * 2.0
	elif t_1 <= 5e-6:
		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 <= -0.999999999998)
		tmp = Float64(Float64(i / alpha) * 2.0);
	elseif (t_1 <= 5e-6)
		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);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.999999999998)
		tmp = (i / alpha) * 2.0;
	elseif (t_1 <= 5e-6)
		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]}, 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, -0.999999999998], N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], 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))) < -0.99999999999800004

   1. Initial program 3.2%

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

   3. Taylor expanded in 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 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 25.5%

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

   if -0.99999999999800004 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.00000000000000041e-6 < (/.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 29.1%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 86.7%

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

Alternative 8: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999999998:\\ \;\;\;\;i \cdot \frac{2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -0.999999999998)
     (* i (/ 2.0 alpha))
     (if (<= t_1 5e-6) 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 <= -0.999999999998) {
		tmp = i * (2.0 / alpha);
	} else if (t_1 <= 5e-6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)
    if (t_1 <= (-0.999999999998d0)) then
        tmp = i * (2.0d0 / alpha)
    else if (t_1 <= 5d-6) 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 t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999999999998) {
		tmp = i * (2.0 / alpha);
	} else if (t_1 <= 5e-6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)
	tmp = 0
	if t_1 <= -0.999999999998:
		tmp = i * (2.0 / alpha)
	elif t_1 <= 5e-6:
		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 <= -0.999999999998)
		tmp = Float64(i * Float64(2.0 / alpha));
	elseif (t_1 <= 5e-6)
		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);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.999999999998)
		tmp = i * (2.0 / alpha);
	elseif (t_1 <= 5e-6)
		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]}, 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, -0.999999999998], N[(i * N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], 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))) < -0.99999999999800004

   1. Initial program 3.2%

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

   3. Taylor expanded in 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 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 25.5%

      \[\leadsto \frac{i}{\alpha} \cdot \color{blue}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.4%

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

   if -0.99999999999800004 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.00000000000000041e-6 < (/.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 29.1%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 86.7%

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

Alternative 9: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 5e-6)
     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)) <= 5e-6) {
		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)) <= 5d-6) 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)) <= 5e-6) {
		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)) <= 5e-6:
		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)) <= 5e-6)
		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)) <= 5e-6)
		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], 5e-6], 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))) < 5.00000000000000041e-6

   1. Initial program 74.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 75.9%

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

   if 5.00000000000000041e-6 < (/.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 29.1%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 86.7%

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

Alternative 10: 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 64.6%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 66.3%

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

Reproduce

herbie shell --seed 2024318 
(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))