Octave 3.8, jcobi/4, as called

Percentage Accurate: 28.3% → 100.0%

Time: 5.5s

Alternatives: 4

Speedup: 25.0×

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: 28.3% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 10000000:\\ \;\;\;\;0.25 \cdot \frac{-i \cdot i}{\mathsf{fma}\left(i \cdot -4, i, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 10000000.0) (* 0.25 (/ (- (* i i)) (fma (* i -4.0) i 1.0))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 10000000.0) {
		tmp = 0.25 * (-(i * i) / fma((i * -4.0), i, 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 10000000.0)
		tmp = Float64(0.25 * Float64(Float64(-Float64(i * i)) / fma(Float64(i * -4.0), i, 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 10000000.0], N[(0.25 * N[((-N[(i * i), $MachinePrecision]) / N[(N[(i * -4.0), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1e7

   1. Initial program 29.3%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. frac-2neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. fma-undefine 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. metadata-eval 100.0%

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

      9. swap-sqr 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. swap-sqr 100.0%

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

      15. metadata-eval 100.0%

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

      16. *-commutative 100.0%

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

      17. distribute-rgt-neg-in 100.0%

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

      18. pow2 100.0%

          \[\leadsto 0.25 \cdot \frac{\left(-i\right) \cdot i}{1 + \color{blue}{{i}^{2}} \cdot \left(-4\right)} \]

      19. metadata-eval 100.0%

          \[\leadsto 0.25 \cdot \frac{\left(-i\right) \cdot i}{1 + {i}^{2} \cdot \color{blue}{-4}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(-i\right) \cdot i}{1 + {i}^{2} \cdot -4}} \]
   7. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto 0.25 \cdot \frac{\left(-i\right) \cdot i}{\color{blue}{{i}^{2} \cdot -4 + 1}} \]

      2. *-commutative 100.0%

         \[\leadsto 0.25 \cdot \frac{\left(-i\right) \cdot i}{\color{blue}{-4 \cdot {i}^{2}} + 1} \]

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 1e7 < i

   1. Initial program 25.1%

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

      1. associate-/l/ 23.5%

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

      2. sub-neg 23.5%

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

      3. swap-sqr 23.5%

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

      4. metadata-eval 23.5%

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

      5. metadata-eval 23.5%

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

      6. swap-sqr 23.5%

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

      7. metadata-eval 23.5%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in i around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 10000000:\\ \;\;\;\;0.25 \cdot \frac{-i \cdot i}{\mathsf{fma}\left(i \cdot -4, i, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 10000000:\\ \;\;\;\;0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 10000000.0) (* 0.25 (* i (/ i (fma i (* i 4.0) -1.0)))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 10000000.0) {
		tmp = 0.25 * (i * (i / fma(i, (i * 4.0), -1.0)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 10000000.0)
		tmp = Float64(0.25 * Float64(i * Float64(i / fma(i, Float64(i * 4.0), -1.0))));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 10000000.0], N[(0.25 * N[(i * N[(i / N[(i * N[(i * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1e7

   1. Initial program 29.3%

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

   3. Simplified 100.0%

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

   if 1e7 < i

   1. Initial program 25.1%

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

      1. associate-/l/ 23.5%

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

      2. sub-neg 23.5%

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

      3. swap-sqr 23.5%

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

      4. metadata-eval 23.5%

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

      5. metadata-eval 23.5%

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

      6. swap-sqr 23.5%

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

      7. metadata-eval 23.5%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in i around inf 100.0%

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

Alternative 3: 98.9% accurate, 2.3× speedup

Math

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

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(Float64(i * i) * Float64(-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[(N[(i * i), $MachinePrecision] * (-0.25)), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 27.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 26.8%

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

      1. associate-/l/ 25.2%

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

      2. sub-neg 25.2%

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

      3. swap-sqr 25.2%

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

      4. metadata-eval 25.2%

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

      5. metadata-eval 25.2%

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

      6. swap-sqr 25.2%

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

      7. metadata-eval 25.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in i around inf 99.3%

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

4. Final simplification 99.1%

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

Alternative 4: 50.4% accurate, 25.0× 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 27.2%

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

   1. associate-/l/ 26.3%

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

   2. sub-neg 26.3%

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

   3. swap-sqr 26.3%

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

   4. metadata-eval 26.3%

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

   5. metadata-eval 26.3%

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

   6. swap-sqr 26.3%

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

   7. metadata-eval 26.3%

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

3. Simplified 26.3%

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

5. Taylor expanded in i around inf 53.6%

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

Reproduce

herbie shell --seed 2024106 
(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)))