Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 84.1%

Time: 25.0s

Alternatives: 7

Speedup: 8.8×

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: 16.3% 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: 84.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t\_2\\ t_4 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_3}{{t\_4}^{2} + -1} \cdot \frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{t\_4}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (/
      (* (/ t_3 (+ (pow t_4 2.0) -1.0)) (/ (fma i t_2 (* alpha beta)) t_4))
      t_4)
     (+
      (+ (/ (* (+ alpha beta) 0.125) i) 0.0625)
      (/ (* (+ alpha beta) -0.125) 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 + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((t_3 / (pow(t_4, 2.0) + -1.0)) * (fma(i, t_2, (alpha * beta)) / t_4)) / t_4;
	} else {
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(t_3 / Float64((t_4 ^ 2.0) + -1.0)) * Float64(fma(i, t_2, Float64(alpha * beta)) / t_4)) / t_4);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) * 0.125) / i) + 0.0625) + Float64(Float64(Float64(alpha + beta) * -0.125) / 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[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$3 / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision] / i), $MachinePrecision]), $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 48.6%

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

      1. associate-/l/ 45.9%

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

      2. times-frac 99.7%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 71.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 71.2%

         \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      2. +-commutative 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      3. fma-define 71.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      4. distribute-lft-out 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      5. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      6. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\color{blue}{\beta + \alpha}}{i}\right) \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      2. associate-/l* 71.2%

         \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \left(\beta + \alpha\right)\right)}{i}} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      4. metadata-eval 71.3%

         \[\leadsto \left(\frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      5. +-commutative 71.3%

         \[\leadsto \left(\frac{0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      6. *-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0.125}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      7. +-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.125}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      8. distribute-lft-neg-in 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      9. metadata-eval 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      10. associate-*r/ 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right)}{i}} \]

      11. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{-0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} \]

      12. *-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\alpha + \beta\right) \cdot -0.125}}{i} \]

      13. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot -0.125}{i} \]

   9. Simplified 71.3%

      \[\leadsto \color{blue}{\left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\beta + \alpha\right) \cdot -0.125}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{{i}^{2} \cdot \frac{{\left(i + \beta\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \end{array} \]

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 (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (* (pow i 2.0) (/ (pow (+ i beta) 2.0) (pow (fma i 2.0 beta) 2.0)))
      t_2)
     (+
      (+ (/ (* (+ alpha beta) 0.125) i) 0.0625)
      (/ (* (+ alpha beta) -0.125) 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 * (i + (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (pow(i, 2.0) * (pow((i + beta), 2.0) / pow(fma(i, 2.0, beta), 2.0))) / t_2;
	} else {
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64((i ^ 2.0) * Float64((Float64(i + beta) ^ 2.0) / (fma(i, 2.0, beta) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) * 0.125) / i) + 0.0625) + Float64(Float64(Float64(alpha + beta) * -0.125) / 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[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[Power[i, 2.0], $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(i * 2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision] / i), $MachinePrecision]), $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 48.6%

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

   3. Taylor expanded in alpha around 0 48.0%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. *-commutative 94.1%

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

      4. fma-undefine 94.1%

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

   5. Simplified 94.1%

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

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 71.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 71.2%

         \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      2. +-commutative 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      3. fma-define 71.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      4. distribute-lft-out 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      5. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      6. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\color{blue}{\beta + \alpha}}{i}\right) \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      2. associate-/l* 71.2%

         \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \left(\beta + \alpha\right)\right)}{i}} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      4. metadata-eval 71.3%

         \[\leadsto \left(\frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      5. +-commutative 71.3%

         \[\leadsto \left(\frac{0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      6. *-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0.125}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      7. +-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.125}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      8. distribute-lft-neg-in 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      9. metadata-eval 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      10. associate-*r/ 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right)}{i}} \]

      11. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{-0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} \]

      12. *-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\alpha + \beta\right) \cdot -0.125}}{i} \]

      13. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot -0.125}{i} \]

   9. Simplified 71.3%

      \[\leadsto \color{blue}{\left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\beta + \alpha\right) \cdot -0.125}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{{i}^{2} \cdot \frac{{\left(i + \beta\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := {\left(\beta + i \cdot 2\right)}^{2}\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(i + \beta\right)}{t\_3 + -1} \cdot \frac{i + \beta}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \end{array} \]

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 (pow (+ beta (* i 2.0)) 2.0)))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (* i (* (/ (* i (+ i beta)) (+ t_3 -1.0)) (/ (+ i beta) t_3)))
     (+
      (+ (/ (* (+ alpha beta) 0.125) i) 0.0625)
      (/ (* (+ alpha beta) -0.125) 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 = pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = i * (((i * (i + beta)) / (t_3 + -1.0)) * ((i + beta) / t_3));
	} else {
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / i);
	}
	return tmp;
}

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 = Math.pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = i * (((i * (i + beta)) / (t_3 + -1.0)) * ((i + beta) / t_3));
	} else {
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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 = math.pow((beta + (i * 2.0)), 2.0)
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= math.inf:
		tmp = i * (((i * (i + beta)) / (t_3 + -1.0)) * ((i + beta) / t_3))
	else:
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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(beta + Float64(i * 2.0)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(i * Float64(Float64(Float64(i * Float64(i + beta)) / Float64(t_3 + -1.0)) * Float64(Float64(i + beta) / t_3)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) * 0.125) / i) + 0.0625) + Float64(Float64(Float64(alpha + beta) * -0.125) / 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 = (beta + (i * 2.0)) ^ 2.0;
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Inf)
		tmp = i * (((i * (i + beta)) / (t_3 + -1.0)) * ((i + beta) / t_3));
	else
		tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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], Infinity], N[(i * N[(N[(N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision] / i), $MachinePrecision]), $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 48.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 93.5%

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

   6. Taylor expanded in alpha around 0 93.4%

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

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 71.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 71.2%

         \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      2. +-commutative 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      3. fma-define 71.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      4. distribute-lft-out 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      5. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      6. +-commutative 71.2%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\color{blue}{\beta + \alpha}}{i}\right) \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 71.2%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      2. associate-/l* 71.2%

         \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \left(\beta + \alpha\right)\right)}{i}} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      4. metadata-eval 71.3%

         \[\leadsto \left(\frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      5. +-commutative 71.3%

         \[\leadsto \left(\frac{0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      6. *-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0.125}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      7. +-commutative 71.3%

         \[\leadsto \left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.125}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      8. distribute-lft-neg-in 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      9. metadata-eval 71.3%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      10. associate-*r/ 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right)}{i}} \]

      11. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{-0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} \]

      12. *-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\alpha + \beta\right) \cdot -0.125}}{i} \]

      13. +-commutative 71.3%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot -0.125}{i} \]

   9. Simplified 71.3%

      \[\leadsto \color{blue}{\left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\beta + \alpha\right) \cdot -0.125}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(i + \beta\right)}{{\left(\beta + i \cdot 2\right)}^{2} + -1} \cdot \frac{i + \beta}{{\left(\beta + i \cdot 2\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \end{array} \]

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) 0.125) i) 0.0625)
      (/ (* (+ alpha beta) -0.125) 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) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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) * 0.125d0) / i) + 0.0625d0) + (((alpha + beta) * (-0.125d0)) / 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) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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(Float64(alpha + beta) * 0.125) / i) + 0.0625) + Float64(Float64(Float64(alpha + beta) * -0.125) / 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) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision] / i), $MachinePrecision]), $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 99.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 0.7%

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

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 73.6%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 73.6%

         \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      2. +-commutative 73.6%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      3. fma-define 73.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      4. distribute-lft-out 73.6%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      5. +-commutative 73.6%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

      6. +-commutative 73.6%

         \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\color{blue}{\beta + \alpha}}{i}\right) \]

   7. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 73.6%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      2. associate-/l* 73.6%

         \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \left(\beta + \alpha\right)\right)}{i}} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      3. associate-*r* 73.7%

         \[\leadsto \left(\frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      4. metadata-eval 73.7%

         \[\leadsto \left(\frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      5. +-commutative 73.7%

         \[\leadsto \left(\frac{0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      6. *-commutative 73.7%

         \[\leadsto \left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0.125}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      7. +-commutative 73.7%

         \[\leadsto \left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.125}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

      8. distribute-lft-neg-in 73.7%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      9. metadata-eval 73.7%

         \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      10. associate-*r/ 73.7%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right)}{i}} \]

      11. +-commutative 73.7%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{-0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} \]

      12. *-commutative 73.7%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\alpha + \beta\right) \cdot -0.125}}{i} \]

      13. +-commutative 73.7%

          \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot -0.125}{i} \]

   9. Simplified 73.7%

      \[\leadsto \color{blue}{\left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\beta + \alpha\right) \cdot -0.125}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (+ (+ (/ (* (+ alpha beta) 0.125) i) 0.0625) (/ (* (+ alpha beta) -0.125) i)))

C

double code(double alpha, double beta, double i) {
	return ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / 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) * 0.125d0) / i) + 0.0625d0) + (((alpha + beta) * (-0.125d0)) / i)
end function

Java

public static double code(double alpha, double beta, double i) {
	return ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / i);
}

Python

def code(alpha, beta, i):
	return ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / i)

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(alpha + beta) * 0.125) / i) + 0.0625) + Float64(Float64(Float64(alpha + beta) * -0.125) / i))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = ((((alpha + beta) * 0.125) / i) + 0.0625) + (((alpha + beta) * -0.125) / i);
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

   1. associate-/l/ 15.6%

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

3. Simplified 15.6%

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

5. Taylor expanded in i around inf 74.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
6. Step-by-step derivation

   1. sub-neg 74.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

   2. +-commutative 74.4%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

   3. fma-define 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

   4. distribute-lft-out 74.4%

      \[\leadsto \mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

   5. +-commutative 74.4%

      \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\alpha + \beta}{i}\right) \]

   6. +-commutative 74.4%

      \[\leadsto \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\color{blue}{\beta + \alpha}}{i}\right) \]

7. Applied egg-rr 74.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
8. Step-by-step derivation

   1. fma-undefine 74.4%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right)} + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   2. associate-/l* 74.4%

      \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \left(\beta + \alpha\right)\right)}{i}} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   3. associate-*r* 74.4%

      \[\leadsto \left(\frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   4. metadata-eval 74.4%

      \[\leadsto \left(\frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   5. +-commutative 74.4%

      \[\leadsto \left(\frac{0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   6. *-commutative 74.4%

      \[\leadsto \left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0.125}}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   7. +-commutative 74.4%

      \[\leadsto \left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.125}{i} + 0.0625\right) + \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \]

   8. distribute-lft-neg-in 74.4%

      \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

   9. metadata-eval 74.4%

      \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

   10. associate-*r/ 74.4%

       \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right)}{i}} \]

   11. +-commutative 74.4%

       \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{-0.125 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i} \]

   12. *-commutative 74.4%

       \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\alpha + \beta\right) \cdot -0.125}}{i} \]

   13. +-commutative 74.4%

       \[\leadsto \left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot -0.125}{i} \]

9. Simplified 74.4%

   \[\leadsto \color{blue}{\left(\frac{\left(\beta + \alpha\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\beta + \alpha\right) \cdot -0.125}{i}} \]

10. Final simplification 74.4%

    \[\leadsto \left(\frac{\left(\alpha + \beta\right) \cdot 0.125}{i} + 0.0625\right) + \frac{\left(\alpha + \beta\right) \cdot -0.125}{i} \]
11. Add Preprocessing

Alternative 6: 72.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+268}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4.8e+268) 0.0625 0.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+268) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (beta <= 4.8d+268) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+268) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.8e+268:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.8e+268)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.8e+268)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.8e+268], 0.0625, 0.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.7999999999999999e268

   1. Initial program 17.8%

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

      1. associate-/l/ 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in i around inf 73.9%

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

   if 4.7999999999999999e268 < 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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 28.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   6. Taylor expanded in i around 0 25.1%

      \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
   7. Step-by-step derivation

      1. div-sub 25.1%

         \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i}} \]

      2. +-commutative 25.1%

         \[\leadsto \frac{0.0625 \cdot \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

      3. distribute-lft-in 25.1%

         \[\leadsto \frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\beta + \alpha\right)\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

      4. associate-*r* 25.3%

         \[\leadsto \frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

      5. metadata-eval 25.3%

         \[\leadsto \frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

      6. associate-*r/ 25.3%

         \[\leadsto \color{blue}{0.125 \cdot \frac{\beta + \alpha}{i}} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

      7. +-commutative 25.3%

         \[\leadsto 0.125 \cdot \frac{\beta + \alpha}{i} - \frac{0.125 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i} \]

      8. associate-*r/ 25.3%

         \[\leadsto 0.125 \cdot \frac{\beta + \alpha}{i} - \color{blue}{0.125 \cdot \frac{\beta + \alpha}{i}} \]

      9. +-inverses 25.3%

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

   8. Simplified 25.3%

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

Alternative 7: 9.6% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0)

C

double code(double alpha, double beta, double i) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0;
}

Python

def code(alpha, beta, i):
	return 0.0

Julia

function code(alpha, beta, i)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 16.5%

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

   1. associate-/l/ 15.6%

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

3. Simplified 15.6%

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

5. Taylor expanded in i around inf 74.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

6. Taylor expanded in i around 0 8.9%

   \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
7. Step-by-step derivation

   1. div-sub 8.9%

      \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i}} \]

   2. +-commutative 8.9%

      \[\leadsto \frac{0.0625 \cdot \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   3. distribute-lft-in 8.9%

      \[\leadsto \frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\beta + \alpha\right)\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   4. associate-*r* 9.0%

      \[\leadsto \frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\beta + \alpha\right)}}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   5. metadata-eval 9.0%

      \[\leadsto \frac{\color{blue}{0.125} \cdot \left(\beta + \alpha\right)}{i} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   6. associate-*r/ 9.0%

      \[\leadsto \color{blue}{0.125 \cdot \frac{\beta + \alpha}{i}} - \frac{0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. +-commutative 9.0%

      \[\leadsto 0.125 \cdot \frac{\beta + \alpha}{i} - \frac{0.125 \cdot \color{blue}{\left(\beta + \alpha\right)}}{i} \]

   8. associate-*r/ 9.0%

      \[\leadsto 0.125 \cdot \frac{\beta + \alpha}{i} - \color{blue}{0.125 \cdot \frac{\beta + \alpha}{i}} \]

   9. +-inverses 9.0%

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

8. Simplified 9.0%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))