Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 84.4%

Time: 25.5s

Alternatives: 6

Speedup: 53.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Initial Program: 16.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Alternative 1: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t\_3\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_4 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{t\_4 \cdot \frac{\mathsf{fma}\left(i, t\_3, \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3)))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       t_4
       (/ (fma i t_3 (* alpha beta)) (pow (fma i 2.0 (+ alpha beta)) 2.0)))
      t_2)
     (/
      (+ (* (+ alpha beta) -0.0625) (+ (* i 0.0625) (* (+ alpha beta) 0.0625)))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i + (alpha + beta);
	double t_4 = i * t_3;
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (t_4 * (fma(i, t_3, (alpha * beta)) / pow(fma(i, 2.0, (alpha + beta)), 2.0))) / t_2;
	} else {
		tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(i + Float64(alpha + beta))
	t_4 = Float64(i * t_3)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(t_4 * Float64(fma(i, t_3, Float64(alpha * beta)) / (fma(i, 2.0, Float64(alpha + beta)) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(alpha + beta) * -0.0625) + Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.0625))) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(t$95$4 * N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] * -0.0625), $MachinePrecision] + N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 43.6%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. pow2 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-undefine 99.7%

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

      12. pow2 99.7%

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

      13. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 7.0%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. associate--l+ 7.0%

         \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(0.25 + \left(0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. *-commutative 7.0%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i} \cdot 0.25} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. distribute-lft-out 7.0%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i} \cdot 0.25 - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative 7.0%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \color{blue}{\frac{\alpha + \beta}{i} \cdot 0.25}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 7.0%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \frac{\alpha + \beta}{i} \cdot 0.25\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around inf 76.5%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 76.5%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. associate-*r/ 76.5%

         \[\leadsto \left(0.0625 + \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      3. distribute-rgt-out-- 76.5%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      4. metadata-eval 76.5%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      5. metadata-eval 76.5%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\alpha + \beta}{i} \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + -0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   9. Taylor expanded in i around 0 76.5%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left(\alpha + \beta\right) + \left(0.0625 \cdot i + 0.0625 \cdot \left(\alpha + \beta\right)\right)}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.1)
     t_3
     (/
      (+ (* (+ alpha beta) -0.0625) (+ (* i 0.0625) (* (+ alpha beta) 0.0625)))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (((alpha + beta) * (-0.0625d0)) + ((i * 0.0625d0) + ((alpha + beta) * 0.0625d0))) / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(Float64(alpha + beta) * -0.0625) + Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.0625))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(N[(N[(alpha + beta), $MachinePrecision] * -0.0625), $MachinePrecision] + N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.5%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.8%

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

   3. Taylor expanded in i around inf 25.2%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. associate--l+ 25.2%

         \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(0.25 + \left(0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. *-commutative 25.2%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i} \cdot 0.25} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. distribute-lft-out 25.2%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i} \cdot 0.25 - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative 25.2%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \color{blue}{\frac{\alpha + \beta}{i} \cdot 0.25}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 25.2%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \frac{\alpha + \beta}{i} \cdot 0.25\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 76.9%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. associate-*r/ 76.9%

         \[\leadsto \left(0.0625 + \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      3. distribute-rgt-out-- 76.9%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      4. metadata-eval 76.9%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      5. metadata-eval 76.9%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\alpha + \beta}{i} \]

   8. Simplified 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + -0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   9. Taylor expanded in i around 0 76.9%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left(\alpha + \beta\right) + \left(0.0625 \cdot i + 0.0625 \cdot \left(\alpha + \beta\right)\right)}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+ (* (+ alpha beta) -0.0625) (+ (* i 0.0625) (* (+ alpha beta) 0.0625)))
  i))

C

double code(double alpha, double beta, double i) {
	return (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((alpha + beta) * (-0.0625d0)) + ((i * 0.0625d0) + ((alpha + beta) * 0.0625d0))) / i
end function

Java

public static double code(double alpha, double beta, double i) {
	return (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
}

Python

def code(alpha, beta, i):
	return (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(alpha + beta) * -0.0625) + Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.0625))) / i)
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((alpha + beta) * -0.0625) + ((i * 0.0625) + ((alpha + beta) * 0.0625))) / i;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(alpha + beta), $MachinePrecision] * -0.0625), $MachinePrecision] + N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in i around inf 33.7%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
4. Step-by-step derivation

   1. associate--l+ 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(0.25 + \left(0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   2. *-commutative 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i} \cdot 0.25} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. distribute-lft-out 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i} \cdot 0.25 - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. *-commutative 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \color{blue}{\frac{\alpha + \beta}{i} \cdot 0.25}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

5. Simplified 33.7%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \frac{\alpha + \beta}{i} \cdot 0.25\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

6. Taylor expanded in i around inf 77.0%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i}} \]

   2. associate-*r/ 77.0%

      \[\leadsto \left(0.0625 + \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   3. distribute-rgt-out-- 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   4. metadata-eval 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   5. metadata-eval 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\alpha + \beta}{i} \]

8. Simplified 77.0%

   \[\leadsto \color{blue}{\left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + -0.0625 \cdot \frac{\alpha + \beta}{i}} \]

9. Taylor expanded in i around 0 77.0%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left(\alpha + \beta\right) + \left(0.0625 \cdot i + 0.0625 \cdot \left(\alpha + \beta\right)\right)}{i}} \]

10. Final simplification 77.0%

    \[\leadsto \frac{\left(\alpha + \beta\right) \cdot -0.0625 + \left(i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.0625\right)}{i} \]
11. Add Preprocessing

Alternative 4: 73.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\alpha + \beta\right) \cdot -0.0625 + 0.0625 \cdot \left(i + \beta\right)}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/ (+ (* (+ alpha beta) -0.0625) (* 0.0625 (+ i beta))) i))

C

double code(double alpha, double beta, double i) {
	return (((alpha + beta) * -0.0625) + (0.0625 * (i + beta))) / i;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((alpha + beta) * (-0.0625d0)) + (0.0625d0 * (i + beta))) / i
end function

Java

public static double code(double alpha, double beta, double i) {
	return (((alpha + beta) * -0.0625) + (0.0625 * (i + beta))) / i;
}

Python

def code(alpha, beta, i):
	return (((alpha + beta) * -0.0625) + (0.0625 * (i + beta))) / i

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(alpha + beta) * -0.0625) + Float64(0.0625 * Float64(i + beta))) / i)
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((alpha + beta) * -0.0625) + (0.0625 * (i + beta))) / i;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(alpha + beta), $MachinePrecision] * -0.0625), $MachinePrecision] + N[(0.0625 * N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in i around inf 33.7%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
4. Step-by-step derivation

   1. associate--l+ 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(0.25 + \left(0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   2. *-commutative 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i} \cdot 0.25} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. distribute-lft-out 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i} \cdot 0.25 - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. *-commutative 33.7%

      \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \color{blue}{\frac{\alpha + \beta}{i} \cdot 0.25}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

5. Simplified 33.7%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \frac{\alpha + \beta}{i} \cdot 0.25\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

6. Taylor expanded in i around inf 77.0%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i}} \]

   2. associate-*r/ 77.0%

      \[\leadsto \left(0.0625 + \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   3. distribute-rgt-out-- 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   4. metadata-eval 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

   5. metadata-eval 77.0%

      \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\alpha + \beta}{i} \]

8. Simplified 77.0%

   \[\leadsto \color{blue}{\left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + -0.0625 \cdot \frac{\alpha + \beta}{i}} \]

9. Taylor expanded in i around 0 77.0%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left(\alpha + \beta\right) + \left(0.0625 \cdot i + 0.0625 \cdot \left(\alpha + \beta\right)\right)}{i}} \]

10. Taylor expanded in alpha around 0 72.6%

    \[\leadsto \frac{-0.0625 \cdot \left(\alpha + \beta\right) + \color{blue}{\left(0.0625 \cdot \beta + 0.0625 \cdot i\right)}}{i} \]
11. Step-by-step derivation

    1. distribute-lft-out 72.6%

       \[\leadsto \frac{-0.0625 \cdot \left(\alpha + \beta\right) + \color{blue}{0.0625 \cdot \left(\beta + i\right)}}{i} \]

12. Simplified 72.6%

    \[\leadsto \frac{-0.0625 \cdot \left(\alpha + \beta\right) + \color{blue}{0.0625 \cdot \left(\beta + i\right)}}{i} \]

13. Final simplification 72.6%

    \[\leadsto \frac{\left(\alpha + \beta\right) \cdot -0.0625 + 0.0625 \cdot \left(i + \beta\right)}{i} \]
14. Add Preprocessing

Alternative 5: 71.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.8e+198) 0.0625 (/ 0.0 i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+198) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	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) :: tmp
    if (beta <= 5.8d+198) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+198) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.8e+198:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.8e+198)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.8e+198)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.8e+198], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.8000000000000002e198

   1. Initial program 18.0%

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

      1. associate-/l/ 15.3%

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

      2. associate-/l* 19.1%

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

      3. +-commutative 19.1%

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

      4. +-commutative 19.1%

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

      5. +-commutative 19.1%

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

      6. associate-+l+ 19.1%

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

      7. +-commutative 19.1%

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

      8. associate-*l* 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in i around inf 76.2%

      \[\leadsto \color{blue}{0.0625} \]

   if 5.8000000000000002e198 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 12.3%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. associate--l+ 12.3%

         \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(0.25 + \left(0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. *-commutative 12.3%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i} \cdot 0.25} - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. distribute-lft-out 12.3%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i} \cdot 0.25 - 0.25 \cdot \frac{\alpha + \beta}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative 12.3%

         \[\leadsto \frac{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \color{blue}{\frac{\alpha + \beta}{i} \cdot 0.25}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 12.3%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(0.25 + \left(\frac{2 \cdot \left(\alpha + \beta\right)}{i} \cdot 0.25 - \frac{\alpha + \beta}{i} \cdot 0.25\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around inf 41.6%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 41.6%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. associate-*r/ 41.6%

         \[\leadsto \left(0.0625 + \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      3. distribute-rgt-out-- 41.6%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      4. metadata-eval 41.6%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i}\right) + \left(-0.0625\right) \cdot \frac{\alpha + \beta}{i} \]

      5. metadata-eval 41.6%

         \[\leadsto \left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\alpha + \beta}{i} \]

   8. Simplified 41.6%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot 0.25\right)}{i}\right) + -0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   9. Taylor expanded in i around 0 23.4%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left(\alpha + \beta\right) + 0.0625 \cdot \left(\alpha + \beta\right)}{i}} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 23.4%

          \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-0.0625 + 0.0625\right)}}{i} \]

       2. metadata-eval 23.4%

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

       3. mul0-rgt 23.4%

          \[\leadsto \frac{\color{blue}{0}}{i} \]

   11. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.7% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.2%

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

   1. associate-/l/ 13.7%

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

   2. associate-/l* 18.0%

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

   3. +-commutative 18.0%

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

   4. +-commutative 18.0%

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

   5. +-commutative 18.0%

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

   6. associate-+l+ 18.0%

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

   7. +-commutative 18.0%

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

   8. associate-*l* 17.9%

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

3. Simplified 17.9%

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

5. Taylor expanded in i around inf 70.7%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))