Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

Alternatives: 5

Speedup: 1.0×

• 33.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x (- 3.0 (* x 2.0)))))

C

double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (3.0d0 - (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Python

def code(x):
	return x * (x * (3.0 - (x * 2.0)))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(3.0 - Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (3.0 - (x * 2.0)));
end

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e-14) (* x (* x 3.0)) (* x (* x -3.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e-14) {
		tmp = x * (x * 3.0);
	} else {
		tmp = x * (x * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d-14) then
        tmp = x * (x * 3.0d0)
    else
        tmp = x * (x * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e-14) {
		tmp = x * (x * 3.0);
	} else {
		tmp = x * (x * -3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e-14:
		tmp = x * (x * 3.0)
	else:
		tmp = x * (x * -3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e-14)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = Float64(x * Float64(x * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e-14)
		tmp = x * (x * 3.0);
	else
		tmp = x * (x * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e-14], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999996e-14

   1. Initial program 99.8%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.9%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 85.6%

      \[\leadsto x \cdot \color{blue}{\left(3 \cdot x\right)} \]

   if 4.59999999999999996e-14 < x

   1. Initial program 99.8%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto x \cdot \color{blue}{\left(\left(3 - x \cdot 2\right) \cdot x\right)} \]

      2. flip-- 99.7%

         \[\leadsto x \cdot \left(\color{blue}{\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + x \cdot 2}} \cdot x\right) \]

      3. *-commutative 99.7%

         \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{2 \cdot x}} \cdot x\right) \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{\left(--2\right)} \cdot x} \cdot x\right) \]

      5. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{3 - -2 \cdot x}} \cdot x\right) \]

      6. *-commutative 99.7%

         \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 - \color{blue}{x \cdot -2}} \cdot x\right) \]

      7. associate-*l/ 99.7%

         \[\leadsto x \cdot \color{blue}{\frac{\left(3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right) \cdot x}{3 - x \cdot -2}} \]

   6. Applied egg-rr 6.5%

      \[\leadsto x \cdot \color{blue}{\frac{\left(9 + {x}^{2} \cdot 4\right) \cdot x}{\mathsf{fma}\left(x, 2, 3\right)}} \]

   7. Taylor expanded in x around 0 6.7%

      \[\leadsto x \cdot \frac{\color{blue}{9 \cdot x}}{\mathsf{fma}\left(x, 2, 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative 6.7%

         \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]

   9. Simplified 6.7%

      \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]
   10. Step-by-step derivation

       1. clear-num 6.7%

          \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot 9}}} \]

       2. un-div-inv 6.7%

          \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot 9}}} \]

       3. *-un-lft-identity 6.7%

          \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(x, 2, 3\right)}}{x \cdot 9}} \]

       4. *-commutative 6.7%

          \[\leadsto \frac{x}{\frac{1 \cdot \mathsf{fma}\left(x, 2, 3\right)}{\color{blue}{9 \cdot x}}} \]

       5. times-frac 6.7%

          \[\leadsto \frac{x}{\color{blue}{\frac{1}{9} \cdot \frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

       6. metadata-eval 6.7%

          \[\leadsto \frac{x}{\color{blue}{0.1111111111111111} \cdot \frac{\mathsf{fma}\left(x, 2, 3\right)}{x}} \]

   11. Applied egg-rr 6.7%

       \[\leadsto \color{blue}{\frac{x}{0.1111111111111111 \cdot \frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]
   12. Step-by-step derivation

       1. associate-/r* 6.7%

          \[\leadsto \color{blue}{\frac{\frac{x}{0.1111111111111111}}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

   13. Simplified 6.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{0.1111111111111111}}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

   14. Taylor expanded in x around inf 6.7%

       \[\leadsto \frac{\frac{x}{0.1111111111111111}}{\color{blue}{2 + 3 \cdot \frac{1}{x}}} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 6.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{0.1111111111111111}}{2 + 3 \cdot \frac{1}{x}}\right)\right)} \]

       2. expm1-undefine 2.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{0.1111111111111111}}{2 + 3 \cdot \frac{1}{x}}\right)} - 1} \]

       3. div-inv 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}}{2 + 3 \cdot \frac{1}{x}}\right)} - 1 \]

       4. metadata-eval 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot \color{blue}{9}}{2 + 3 \cdot \frac{1}{x}}\right)} - 1 \]

       5. div-inv 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot 9}{2 + \color{blue}{\frac{3}{x}}}\right)} - 1 \]

   16. Applied egg-rr 2.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 9}{2 + \frac{3}{x}}\right)} - 1} \]
   17. Step-by-step derivation

       1. log1p-undefine 2.5%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{x \cdot 9}{2 + \frac{3}{x}}\right)}} - 1 \]

       2. rem-exp-log 2.5%

          \[\leadsto \color{blue}{\left(1 + \frac{x \cdot 9}{2 + \frac{3}{x}}\right)} - 1 \]

       3. +-commutative 2.5%

          \[\leadsto \color{blue}{\left(\frac{x \cdot 9}{2 + \frac{3}{x}} + 1\right)} - 1 \]

       4. associate--l+ 6.7%

          \[\leadsto \color{blue}{\frac{x \cdot 9}{2 + \frac{3}{x}} + \left(1 - 1\right)} \]

       5. metadata-eval 6.7%

          \[\leadsto \frac{x \cdot 9}{2 + \frac{3}{x}} + \color{blue}{0} \]

       6. +-commutative 6.7%

          \[\leadsto \color{blue}{0 + \frac{x \cdot 9}{2 + \frac{3}{x}}} \]

       7. +-lft-identity 6.7%

          \[\leadsto \color{blue}{\frac{x \cdot 9}{2 + \frac{3}{x}}} \]

       8. *-commutative 6.7%

          \[\leadsto \frac{\color{blue}{9 \cdot x}}{2 + \frac{3}{x}} \]

       9. associate-/l* 6.7%

          \[\leadsto \color{blue}{9 \cdot \frac{x}{2 + \frac{3}{x}}} \]

       10. metadata-eval 6.7%

           \[\leadsto \color{blue}{\frac{1}{0.1111111111111111}} \cdot \frac{x}{2 + \frac{3}{x}} \]

       11. times-frac 6.7%

           \[\leadsto \color{blue}{\frac{1 \cdot x}{0.1111111111111111 \cdot \left(2 + \frac{3}{x}\right)}} \]

       12. *-lft-identity 6.7%

           \[\leadsto \frac{\color{blue}{x}}{0.1111111111111111 \cdot \left(2 + \frac{3}{x}\right)} \]

       13. +-commutative 6.7%

           \[\leadsto \frac{x}{0.1111111111111111 \cdot \color{blue}{\left(\frac{3}{x} + 2\right)}} \]

       14. distribute-lft-in 6.7%

           \[\leadsto \frac{x}{\color{blue}{0.1111111111111111 \cdot \frac{3}{x} + 0.1111111111111111 \cdot 2}} \]

       15. associate-*r/ 6.7%

           \[\leadsto \frac{x}{\color{blue}{\frac{0.1111111111111111 \cdot 3}{x}} + 0.1111111111111111 \cdot 2} \]

       16. metadata-eval 6.7%

           \[\leadsto \frac{x}{\frac{\color{blue}{0.3333333333333333}}{x} + 0.1111111111111111 \cdot 2} \]

       17. metadata-eval 6.7%

           \[\leadsto \frac{x}{\frac{0.3333333333333333}{x} + \color{blue}{0.2222222222222222}} \]

   18. Simplified 6.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{0.3333333333333333}{x} + 0.2222222222222222}} \]

   19. Taylor expanded in x around 0 5.0%

       \[\leadsto \frac{x}{\color{blue}{\frac{0.3333333333333333}{x}}} \]
   20. Step-by-step derivation

       1. add-sqr-sqrt 5.0%

          \[\leadsto \frac{x}{\frac{0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       2. sqrt-prod 5.0%

          \[\leadsto \frac{x}{\frac{0.3333333333333333}{\color{blue}{\sqrt{x \cdot x}}}} \]

       3. sqr-neg 5.0%

          \[\leadsto \frac{x}{\frac{0.3333333333333333}{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}} \]

       4. sqrt-unprod 0.0%

          \[\leadsto \frac{x}{\frac{0.3333333333333333}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

       5. add-sqr-sqrt 39.8%

          \[\leadsto \frac{x}{\frac{0.3333333333333333}{\color{blue}{-x}}} \]

       6. distribute-neg-frac2 39.8%

          \[\leadsto \frac{x}{\color{blue}{-\frac{0.3333333333333333}{x}}} \]

       7. div-inv 39.8%

          \[\leadsto \frac{x}{-\color{blue}{0.3333333333333333 \cdot \frac{1}{x}}} \]

       8. distribute-lft-neg-in 39.8%

          \[\leadsto \frac{x}{\color{blue}{\left(-0.3333333333333333\right) \cdot \frac{1}{x}}} \]

       9. metadata-eval 39.8%

          \[\leadsto \frac{x}{\color{blue}{-0.3333333333333333} \cdot \frac{1}{x}} \]

   21. Applied egg-rr 39.8%

       \[\leadsto \frac{x}{\color{blue}{-0.3333333333333333 \cdot \frac{1}{x}}} \]
   22. Step-by-step derivation

       1. un-div-inv 39.8%

          \[\leadsto \frac{x}{\color{blue}{\frac{-0.3333333333333333}{x}}} \]

       2. associate-/r/ 39.8%

          \[\leadsto \color{blue}{\frac{x}{-0.3333333333333333} \cdot x} \]

       3. div-inv 39.8%

          \[\leadsto \color{blue}{\left(x \cdot \frac{1}{-0.3333333333333333}\right)} \cdot x \]

       4. metadata-eval 39.8%

          \[\leadsto \left(x \cdot \color{blue}{-3}\right) \cdot x \]

   23. Applied egg-rr 39.8%

       \[\leadsto \color{blue}{\left(x \cdot -3\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 3.0)))

C

double code(double x) {
	return x * (x * 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 3.0d0)
end function

Java

public static double code(double x) {
	return x * (x * 3.0);
}

Python

def code(x):
	return x * (x * 3.0)

Julia

function code(x)
	return Float64(x * Float64(x * 3.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 3.0);
end

Wolfram

code[x_] := N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 64.2%

   \[\leadsto x \cdot \color{blue}{\left(3 \cdot x\right)} \]

6. Final simplification 64.2%

   \[\leadsto x \cdot \left(x \cdot 3\right) \]
7. Add Preprocessing

Alternative 4: 3.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 4.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 4.5))

C

double code(double x) {
	return x * 4.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 4.5d0
end function

Java

public static double code(double x) {
	return x * 4.5;
}

Python

def code(x):
	return x * 4.5

Julia

function code(x)
	return Float64(x * 4.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 4.5;
end

Wolfram

code[x_] := N[(x * 4.5), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto x \cdot \color{blue}{\left(\left(3 - x \cdot 2\right) \cdot x\right)} \]

   2. flip-- 99.8%

      \[\leadsto x \cdot \left(\color{blue}{\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + x \cdot 2}} \cdot x\right) \]

   3. *-commutative 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{2 \cdot x}} \cdot x\right) \]

   4. metadata-eval 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{\left(--2\right)} \cdot x} \cdot x\right) \]

   5. cancel-sign-sub-inv 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{3 - -2 \cdot x}} \cdot x\right) \]

   6. *-commutative 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 - \color{blue}{x \cdot -2}} \cdot x\right) \]

   7. associate-*l/ 99.8%

      \[\leadsto x \cdot \color{blue}{\frac{\left(3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right) \cdot x}{3 - x \cdot -2}} \]

6. Applied egg-rr 53.6%

   \[\leadsto x \cdot \color{blue}{\frac{\left(9 + {x}^{2} \cdot 4\right) \cdot x}{\mathsf{fma}\left(x, 2, 3\right)}} \]

7. Taylor expanded in x around 0 53.7%

   \[\leadsto x \cdot \frac{\color{blue}{9 \cdot x}}{\mathsf{fma}\left(x, 2, 3\right)} \]
8. Step-by-step derivation

   1. *-commutative 53.7%

      \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]

9. Simplified 53.7%

   \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]

10. Taylor expanded in x around inf 3.5%

    \[\leadsto x \cdot \color{blue}{4.5} \]
11. Add Preprocessing

Alternative 5: 2.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ -6.75 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -6.75)

C

double code(double x) {
	return -6.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -6.75d0
end function

Java

public static double code(double x) {
	return -6.75;
}

Python

def code(x):
	return -6.75

Julia

function code(x)
	return -6.75
end

MATLAB

function tmp = code(x)
	tmp = -6.75;
end

Wolfram

code[x_] := -6.75

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto x \cdot \color{blue}{\left(\left(3 - x \cdot 2\right) \cdot x\right)} \]

   2. flip-- 99.8%

      \[\leadsto x \cdot \left(\color{blue}{\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + x \cdot 2}} \cdot x\right) \]

   3. *-commutative 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{2 \cdot x}} \cdot x\right) \]

   4. metadata-eval 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 + \color{blue}{\left(--2\right)} \cdot x} \cdot x\right) \]

   5. cancel-sign-sub-inv 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{3 - -2 \cdot x}} \cdot x\right) \]

   6. *-commutative 99.8%

      \[\leadsto x \cdot \left(\frac{3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{3 - \color{blue}{x \cdot -2}} \cdot x\right) \]

   7. associate-*l/ 99.8%

      \[\leadsto x \cdot \color{blue}{\frac{\left(3 \cdot 3 - \left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right) \cdot x}{3 - x \cdot -2}} \]

6. Applied egg-rr 53.6%

   \[\leadsto x \cdot \color{blue}{\frac{\left(9 + {x}^{2} \cdot 4\right) \cdot x}{\mathsf{fma}\left(x, 2, 3\right)}} \]

7. Taylor expanded in x around 0 53.7%

   \[\leadsto x \cdot \frac{\color{blue}{9 \cdot x}}{\mathsf{fma}\left(x, 2, 3\right)} \]
8. Step-by-step derivation

   1. *-commutative 53.7%

      \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]

9. Simplified 53.7%

   \[\leadsto x \cdot \frac{\color{blue}{x \cdot 9}}{\mathsf{fma}\left(x, 2, 3\right)} \]

10. Taylor expanded in x around inf 1.7%

    \[\leadsto x \cdot \color{blue}{\left(4.5 - 6.75 \cdot \frac{1}{x}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 1.7%

       \[\leadsto x \cdot \left(4.5 - \color{blue}{\frac{6.75 \cdot 1}{x}}\right) \]

    2. metadata-eval 1.7%

       \[\leadsto x \cdot \left(4.5 - \frac{\color{blue}{6.75}}{x}\right) \]

12. Simplified 1.7%

    \[\leadsto x \cdot \color{blue}{\left(4.5 - \frac{6.75}{x}\right)} \]

13. Taylor expanded in x around 0 2.7%

    \[\leadsto \color{blue}{-6.75} \]
14. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x (- 3.0 (* x 2.0)))))

C

double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (3.0d0 - (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Python

def code(x):
	return x * (x * (3.0 - (x * 2.0)))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(3.0 - Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (3.0 - (x * 2.0)));
end

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024105 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :alt
  (* x (* x (- 3.0 (* x 2.0))))

  (* (* x x) (- 3.0 (* x 2.0))))