Octave 3.8, jcobi/4, as called

Percentage Accurate: 20.3% → 63.7%

Time: 2.1s

Alternatives: 6

Speedup: 38.2×

• Could not converge on a ground truth

  1. i = -1.7030234586554808e+258

Specification

\[i > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

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

Initial Program: 20.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

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

Alternative 1: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|i\right| \leq 110:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(2, \left|i\right|, -1\right)}{\frac{\left(\left(\left|i\right| \cdot \left|i\right|\right) \cdot 0.25\right) \cdot 1}{\mathsf{fma}\left(2, \left|i\right|, 1\right)}}}\\ \mathbf{elif}\;\left|i\right| \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(0 \cdot 0\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
  :precision binary64
  (if (<= (fabs i) 110.0)
  (/
   1.0
   (/
    (fma 2.0 (fabs i) -1.0)
    (/
     (* (* (* (fabs i) (fabs i)) 0.25) 1.0)
     (fma 2.0 (fabs i) 1.0))))
  (if (<= (fabs i) 4e+133) (* (* 0.0 0.0) -0.25) 0.0625)))

C

double code(double i) {
	double tmp;
	if (fabs(i) <= 110.0) {
		tmp = 1.0 / (fma(2.0, fabs(i), -1.0) / ((((fabs(i) * fabs(i)) * 0.25) * 1.0) / fma(2.0, fabs(i), 1.0)));
	} else if (fabs(i) <= 4e+133) {
		tmp = (0.0 * 0.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (abs(i) <= 110.0)
		tmp = Float64(1.0 / Float64(fma(2.0, abs(i), -1.0) / Float64(Float64(Float64(Float64(abs(i) * abs(i)) * 0.25) * 1.0) / fma(2.0, abs(i), 1.0))));
	elseif (abs(i) <= 4e+133)
		tmp = Float64(Float64(0.0 * 0.0) * -0.25);
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[N[Abs[i], $MachinePrecision], 110.0], N[(1.0 / N[(N[(2.0 * N[Abs[i], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[(N[(N[Abs[i], $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * 1.0), $MachinePrecision] / N[(2.0 * N[Abs[i], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[i], $MachinePrecision], 4e+133], N[(N[(0.0 * 0.0), $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if i < 110

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-sqr-1 N/A

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

      5. associate-/r* N/A

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

      6. div-flip N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. remove-sound-/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(2, i, -1\right)}{\frac{\left(\left(i \cdot i\right) \cdot 0.25\right) \cdot 1}{\mathsf{fma}\left(2, i, 1\right)}}}} \]

   if 110 < i < 4.0000000000000001e133

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]
   8. Step-by-step derivation

   9. Applied rewrites 64.1%

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]

   if 4.0000000000000001e133 < i

   1. Initial program 20.3%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 63.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|i\right| \leq 110:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(4 \cdot \left|i\right|, \left|i\right|, -1\right)}{0.25 \cdot \left(\left|i\right| \cdot \left|i\right|\right)}}\\ \mathbf{elif}\;\left|i\right| \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(0 \cdot 0\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
  :precision binary64
  (if (<= (fabs i) 110.0)
  (/
   1.0
   (/
    (fma (* 4.0 (fabs i)) (fabs i) -1.0)
    (* 0.25 (* (fabs i) (fabs i)))))
  (if (<= (fabs i) 4e+133) (* (* 0.0 0.0) -0.25) 0.0625)))

C

double code(double i) {
	double tmp;
	if (fabs(i) <= 110.0) {
		tmp = 1.0 / (fma((4.0 * fabs(i)), fabs(i), -1.0) / (0.25 * (fabs(i) * fabs(i))));
	} else if (fabs(i) <= 4e+133) {
		tmp = (0.0 * 0.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (abs(i) <= 110.0)
		tmp = Float64(1.0 / Float64(fma(Float64(4.0 * abs(i)), abs(i), -1.0) / Float64(0.25 * Float64(abs(i) * abs(i)))));
	elseif (abs(i) <= 4e+133)
		tmp = Float64(Float64(0.0 * 0.0) * -0.25);
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[N[Abs[i], $MachinePrecision], 110.0], N[(1.0 / N[(N[(N[(4.0 * N[Abs[i], $MachinePrecision]), $MachinePrecision] * N[Abs[i], $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.25 * N[(N[Abs[i], $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[i], $MachinePrecision], 4e+133], N[(N[(0.0 * 0.0), $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if i < 110

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 64.3%

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

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4 \cdot i, i, -1\right)}{0.25 \cdot \left(i \cdot i\right)}}} \]

   if 110 < i < 4.0000000000000001e133

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]
   8. Step-by-step derivation

   9. Applied rewrites 64.1%

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]

   if 4.0000000000000001e133 < i

   1. Initial program 20.3%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|i\right| \leq 110:\\ \;\;\;\;\frac{\left|i\right|}{\mathsf{fma}\left(16, \left|i\right| \cdot \left|i\right|, -4\right)} \cdot \left|i\right|\\ \mathbf{elif}\;\left|i\right| \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(0 \cdot 0\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
  :precision binary64
  (if (<= (fabs i) 110.0)
  (* (/ (fabs i) (fma 16.0 (* (fabs i) (fabs i)) -4.0)) (fabs i))
  (if (<= (fabs i) 4e+133) (* (* 0.0 0.0) -0.25) 0.0625)))

C

double code(double i) {
	double tmp;
	if (fabs(i) <= 110.0) {
		tmp = (fabs(i) / fma(16.0, (fabs(i) * fabs(i)), -4.0)) * fabs(i);
	} else if (fabs(i) <= 4e+133) {
		tmp = (0.0 * 0.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (abs(i) <= 110.0)
		tmp = Float64(Float64(abs(i) / fma(16.0, Float64(abs(i) * abs(i)), -4.0)) * abs(i));
	elseif (abs(i) <= 4e+133)
		tmp = Float64(Float64(0.0 * 0.0) * -0.25);
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[N[Abs[i], $MachinePrecision], 110.0], N[(N[(N[Abs[i], $MachinePrecision] / N[(16.0 * N[(N[Abs[i], $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[i], $MachinePrecision], 4e+133], N[(N[(0.0 * 0.0), $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if i < 110

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 64.3%

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\left(\frac{0.25}{\mathsf{fma}\left(4 \cdot i, i, -1\right)} \cdot i\right) \cdot i} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-out N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. mult-flip N/A

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

   7. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(16, i \cdot i, -4\right)}} \cdot i \]

   if 110 < i < 4.0000000000000001e133

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]
   8. Step-by-step derivation

   9. Applied rewrites 64.1%

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]

   if 4.0000000000000001e133 < i

   1. Initial program 20.3%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|i\right| \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\left(\left|i\right| \cdot \left|i\right|\right) \cdot -0.25\\ \mathbf{elif}\;\left|i\right| \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(0 \cdot 0\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
  :precision binary64
  (if (<= (fabs i) 7e-9)
  (* (* (fabs i) (fabs i)) -0.25)
  (if (<= (fabs i) 4e+133) (* (* 0.0 0.0) -0.25) 0.0625)))

C

double code(double i) {
	double tmp;
	if (fabs(i) <= 7e-9) {
		tmp = (fabs(i) * fabs(i)) * -0.25;
	} else if (fabs(i) <= 4e+133) {
		tmp = (0.0 * 0.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8) :: tmp
    if (abs(i) <= 7d-9) then
        tmp = (abs(i) * abs(i)) * (-0.25d0)
    else if (abs(i) <= 4d+133) then
        tmp = (0.0d0 * 0.0d0) * (-0.25d0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (Math.abs(i) <= 7e-9) {
		tmp = (Math.abs(i) * Math.abs(i)) * -0.25;
	} else if (Math.abs(i) <= 4e+133) {
		tmp = (0.0 * 0.0) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if math.fabs(i) <= 7e-9:
		tmp = (math.fabs(i) * math.fabs(i)) * -0.25
	elif math.fabs(i) <= 4e+133:
		tmp = (0.0 * 0.0) * -0.25
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (abs(i) <= 7e-9)
		tmp = Float64(Float64(abs(i) * abs(i)) * -0.25);
	elseif (abs(i) <= 4e+133)
		tmp = Float64(Float64(0.0 * 0.0) * -0.25);
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (abs(i) <= 7e-9)
		tmp = (abs(i) * abs(i)) * -0.25;
	elseif (abs(i) <= 4e+133)
		tmp = (0.0 * 0.0) * -0.25;
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[N[Abs[i], $MachinePrecision], 7e-9], N[(N[(N[Abs[i], $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], If[LessEqual[N[Abs[i], $MachinePrecision], 4e+133], N[(N[(0.0 * 0.0), $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if i < 6.9999999999999998e-9

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   if 6.9999999999999998e-9 < i < 4.0000000000000001e133

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]
   8. Step-by-step derivation

   9. Applied rewrites 64.1%

      \[\leadsto \left(\color{blue}{0} \cdot i\right) \cdot -0.25 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(0 \cdot \color{blue}{0}\right) \cdot -0.25 \]

   if 4.0000000000000001e133 < i

   1. Initial program 20.3%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|i\right| \leq 0.023:\\ \;\;\;\;\left(\left|i\right| \cdot \left|i\right|\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (i)
  :precision binary64
  (if (<= (fabs i) 0.023) (* (* (fabs i) (fabs i)) -0.25) 0.0625))

C

double code(double i) {
	double tmp;
	if (fabs(i) <= 0.023) {
		tmp = (fabs(i) * fabs(i)) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8) :: tmp
    if (abs(i) <= 0.023d0) then
        tmp = (abs(i) * abs(i)) * (-0.25d0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (Math.abs(i) <= 0.023) {
		tmp = (Math.abs(i) * Math.abs(i)) * -0.25;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if math.fabs(i) <= 0.023:
		tmp = (math.fabs(i) * math.fabs(i)) * -0.25
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (abs(i) <= 0.023)
		tmp = Float64(Float64(abs(i) * abs(i)) * -0.25);
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (abs(i) <= 0.023)
		tmp = (abs(i) * abs(i)) * -0.25;
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[N[Abs[i], $MachinePrecision], 0.023], N[(N[(N[Abs[i], $MachinePrecision] * N[Abs[i], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.023

   1. Initial program 20.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 39.0%

      \[\leadsto \left(i \cdot i\right) \cdot \color{blue}{-0.25} \]

   if 0.023 < i

   1. Initial program 20.3%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 26.7% accurate, 38.2× speedup

Math

\[0.0625 \]

FPCore

(FPCore (i)
  :precision binary64
  0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
use fmin_fmax_functions
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

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

Python

def code(i):
	return 0.0625

Julia

function code(i)
	return 0.0625
end

MATLAB

function tmp = code(i)
	tmp = 0.0625;
end

Wolfram

code[i_] := 0.0625

Derivation

1. Initial program 20.3%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
3. Step-by-step derivation

4. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))