Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.1% → 99.2%

Time: 8.2s

Alternatives: 5

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: 27.1% 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: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(i \cdot \left(0 - i \cdot i\right), i, i \cdot \left(i \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (fma (* i (- 0.0 (* i i))) i (* i (* i -0.25))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = fma((i * (0.0 - (i * i))), i, (i * (i * -0.25)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

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

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(N[(i * N[(0.0 - N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + N[(i * N[(i * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 25.5%

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

   3. Taylor expanded in i around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{i}^{2} \cdot \left(-1 \cdot {i}^{2} - \frac{1}{4}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({i}^{2}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right)} \]

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{i}^{2} + 0}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{i \cdot i} + 0, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, i, 0\right)}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{-1 \cdot {i}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 0\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), -1 \cdot {i}^{2} + \color{blue}{\frac{-1}{4}}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} + -1 \cdot {i}^{2}}, 0\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} + \color{blue}{\left(\mathsf{neg}\left({i}^{2}\right)\right)}, 0\right) \]

      10. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} - {i}^{2}}, 0\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} - {i}^{2}}, 0\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} - \color{blue}{\left({i}^{2} + 0\right)}, 0\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} - \left(\color{blue}{i \cdot i} + 0\right), 0\right) \]

      14. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), -0.25 - \mathsf{fma}\left(i, i, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(i \cdot i + 0\right) \cdot \left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(i \cdot i\right)} \cdot \left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot \left(i \cdot i\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right) \cdot i} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right) \cdot i} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right)} \cdot i \]

      7. +-rgt-identity N/A

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

      8. --lowering--.f64 N/A

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

      9. +-rgt-identity N/A

         \[\leadsto \left(\left(\frac{-1}{4} - \color{blue}{\left(i \cdot i + 0\right)}\right) \cdot i\right) \cdot i \]

      10. accelerator-lowering-fma.f64 99.4

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

   7. Applied egg-rr 99.4%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.4

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

   9. Applied egg-rr 99.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. sub-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*r* N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

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

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

       9. pow3 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{{i}^{3}}\right), i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       10. sqr-pow N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{{i}^{\left(\frac{3}{2}\right)} \cdot {i}^{\left(\frac{3}{2}\right)}}\right), i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       11. unpow-prod-down N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{{\left(i \cdot i\right)}^{\left(\frac{3}{2}\right)}}\right), i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left({\left(i \cdot i\right)}^{\left(\frac{3}{2}\right)}\right)}, i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       13. unpow-prod-down N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{{i}^{\left(\frac{3}{2}\right)} \cdot {i}^{\left(\frac{3}{2}\right)}}\right), i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       14. sqr-pow N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{{i}^{3}}\right), i, \frac{-1}{4} \cdot \left(i \cdot i\right)\right) \]

       15. cube-unmult N/A

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

       16. *-lowering-*.f64 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

       20. *-lowering-*.f64 N/A

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

       21. *-lowering-*.f64 99.4

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

   11. Applied egg-rr 99.4%

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

   if 0.5 < i

   1. Initial program 21.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(i \cdot \left(0 - i \cdot i\right), i, i \cdot \left(i \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.3%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. clear-num N/A

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

   10. div-inv N/A

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

4. Applied egg-rr 99.6%

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

Alternative 3: 99.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = 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(i * Float64(i * 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 = i * (i * (-0.25 - (i * i)));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 25.5%

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

   3. Taylor expanded in i around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{i}^{2} \cdot \left(-1 \cdot {i}^{2} - \frac{1}{4}\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({i}^{2}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right)} \]

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{i}^{2} + 0}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{i \cdot i} + 0, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, i, 0\right)}, -1 \cdot {i}^{2} - \frac{1}{4}, 0\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{-1 \cdot {i}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 0\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), -1 \cdot {i}^{2} + \color{blue}{\frac{-1}{4}}, 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} + -1 \cdot {i}^{2}}, 0\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} + \color{blue}{\left(\mathsf{neg}\left({i}^{2}\right)\right)}, 0\right) \]

      10. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} - {i}^{2}}, 0\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \color{blue}{\frac{-1}{4} - {i}^{2}}, 0\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} - \color{blue}{\left({i}^{2} + 0\right)}, 0\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), \frac{-1}{4} - \left(\color{blue}{i \cdot i} + 0\right), 0\right) \]

      14. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, i, 0\right), -0.25 - \mathsf{fma}\left(i, i, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(i \cdot i + 0\right) \cdot \left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(i \cdot i\right)} \cdot \left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot \left(i \cdot i\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right) \cdot i} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right) \cdot i} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} - \left(i \cdot i + 0\right)\right) \cdot i\right)} \cdot i \]

      7. +-rgt-identity N/A

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

      8. --lowering--.f64 N/A

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

      9. +-rgt-identity N/A

         \[\leadsto \left(\left(\frac{-1}{4} - \color{blue}{\left(i \cdot i + 0\right)}\right) \cdot i\right) \cdot i \]

      10. accelerator-lowering-fma.f64 99.4

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

   7. Applied egg-rr 99.4%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.4

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

   9. Applied egg-rr 99.4%

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

   if 0.5 < i

   1. Initial program 21.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

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

Alternative 4: 99.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} 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 = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 25.5%

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

   3. Taylor expanded in i around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot {i}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(i \cdot i\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot i\right) \cdot i} + 0 \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(\frac{-1}{4} \cdot i\right)} + 0 \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, \frac{-1}{4} \cdot i, 0\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\frac{-1}{4} \cdot i + 0}, 0\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{-1}{4}} + 0, 0\right) \]

      8. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, -0.25, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{-1}{4}}, 0\right) \]

      2. *-lowering-*.f64 98.8

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

   7. Applied egg-rr 98.8%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.8

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

   9. Applied egg-rr 98.8%

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

   if 0.5 < i

   1. Initial program 21.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.4%

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

Alternative 5: 51.7% 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 23.3%

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

3. Taylor expanded in i around inf

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

5. Simplified 51.7%

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

Reproduce

herbie shell --seed 2024199 
(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)))