Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.5% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 71.0×

Specification

\[i > 0\]

Math

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

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)
    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: 27.5% accurate, 1.0× speedup

Math

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

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)
    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: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 20000000:\\ \;\;\;\;\frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, 16, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 20000000.0) (/ (* i i) (fma (* i i) 16.0 -4.0)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 20000000.0) {
		tmp = (i * i) / fma((i * i), 16.0, -4.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 20000000.0)
		tmp = Float64(Float64(i * i) / fma(Float64(i * i), 16.0, -4.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 20000000.0], N[(N[(i * i), $MachinePrecision] / N[(N[(i * i), $MachinePrecision] * 16.0 + -4.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 2e7

   1. Initial program 41.6%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. 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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. swap-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. swap-sqr N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{i \cdot i}{\left(i \cdot i\right) \cdot \color{blue}{16} + -1 \cdot 4} \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, \color{blue}{16}, 4 \cdot -1\right)} \]

      22. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   if 2e7 < i

   1. Initial program 27.5%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(-i\right) \cdot \mathsf{fma}\left(i \cdot i, i, 0.25 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5)
   (* (- i) (fma (* i i) i (* 0.25 i)))
   (+ (/ 0.015625 (* i i)) 0.0625)))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = -i * fma((i * i), i, (0.25 * i));
	} else {
		tmp = (0.015625 / (i * i)) + 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(-i) * fma(Float64(i * i), i, Float64(0.25 * i)));
	else
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[((-i) * N[(N[(i * i), $MachinePrecision] * i + N[(0.25 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 39.0%

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

   3. Taylor expanded in i around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 0.5 < i

   1. Initial program 31.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

      7. lower-*.f64 97.8

         \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(-i\right) \cdot \mathsf{fma}\left(i \cdot i, i, 0.25 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(i, i, 0.25\right) \cdot i\right) \cdot \left(-i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5)
   (* (* (fma i i 0.25) i) (- i))
   (+ (/ 0.015625 (* i i)) 0.0625)))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (fma(i, i, 0.25) * i) * -i;
	} else {
		tmp = (0.015625 / (i * i)) + 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(fma(i, i, 0.25) * i) * Float64(-i));
	else
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(N[(N[(i * i + 0.25), $MachinePrecision] * i), $MachinePrecision] * (-i)), $MachinePrecision], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 39.0%

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

   3. Taylor expanded in i around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.5 < i

   1. Initial program 31.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

      7. lower-*.f64 97.8

         \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(i, i, 0.25\right) \cdot i\right) \cdot \left(-i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* (* (fma i i 0.25) i) (- i)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (fma(i, i, 0.25) * i) * -i;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(fma(i, i, 0.25) * i) * Float64(-i));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(N[(N[(i * i + 0.25), $MachinePrecision] * i), $MachinePrecision] * (-i)), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 39.0%

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

   3. Taylor expanded in i around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.5 < i

   1. Initial program 31.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.0%

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

Alternative 5: 98.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;-0.25 \cdot \left(i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* -0.25 (* i i)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = -0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = (-0.25d0) * (i * i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = -0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = -0.25 * (i * i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(-0.25 * Float64(i * i));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = -0.25 * (i * i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(-0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 39.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot {i}^{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.5 < i

   1. Initial program 31.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.0%

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

Alternative 6: 52.1% accurate, 71.0× speedup

Math

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

FPCore

(FPCore (i) :precision binary64 0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
    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 35.1%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 48.0%

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

Reproduce

herbie shell --seed 2024270 
(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)))