Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.6% → 100.0%

Time: 7.1s

Alternatives: 8

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: 27.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 46000000:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(16, i \cdot i, -4\right)} \cdot i\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 46000000.0) (* (/ i (fma 16.0 (* i i) -4.0)) i) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 46000000.0) {
		tmp = (i / fma(16.0, (i * i), -4.0)) * i;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 46000000.0)
		tmp = Float64(Float64(i / fma(16.0, Float64(i * i), -4.0)) * i);
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 46000000.0], N[(N[(i / N[(16.0 * N[(i * i), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 4.6e7

   1. Initial program 30.6%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

   if 4.6e7 < i

   1. Initial program 20.2%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{16 - \frac{4}{i \cdot i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 1.0 (- 16.0 (/ 4.0 (* i i)))))

C

double code(double i) {
	return 1.0 / (16.0 - (4.0 / (i * i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 1.0d0 / (16.0d0 - (4.0d0 / (i * i)))
end function

Java

public static double code(double i) {
	return 1.0 / (16.0 - (4.0 / (i * i)));
}

Python

def code(i):
	return 1.0 / (16.0 - (4.0 / (i * i)))

Julia

function code(i)
	return Float64(1.0 / Float64(16.0 - Float64(4.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = 1.0 / (16.0 - (4.0 / (i * i)));
end

Wolfram

code[i_] := N[(1.0 / N[(16.0 - N[(4.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.4%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{16 - \frac{4}{i \cdot i}}} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = fma(i, -0.25, ((-i * i) * 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(fma(i, -0.25, Float64(Float64(Float64(-i) * i) * i)) * 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[(i * -0.25 + N[(N[((-i) * i), $MachinePrecision] * i), $MachinePrecision]), $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 28.9%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.2%

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

   if 0.5 < i

   1. Initial program 22.0%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 4: 99.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5)
   (* (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(fma(i, i, 0.25) * Float64(Float64(-i) * 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[(i * i + 0.25), $MachinePrecision] * N[((-i) * i), $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 28.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. unsub-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

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

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   if 0.5 < i

   1. Initial program 22.0%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 99.1% accurate, 2.8× speedup

Math

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

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(fma(i, i, 0.25) * Float64(Float64(-i) * i));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. unsub-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

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

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   if 0.5 < i

   1. Initial program 22.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.0%

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

Alternative 6: 99.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(Float64(-0.25 - Float64(i * i)) * 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)) * i) * i;
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.9%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.5 < i

   1. Initial program 22.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.0%

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

Alternative 7: 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 28.9%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.5 < i

   1. Initial program 22.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.0%

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

Alternative 8: 51.2% 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 25.4%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2024235 
(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)))