Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 6

Speedup: 1.0×

• 33.7% 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: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\frac{x}{\frac{0.1111111111111111}{x \cdot 0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5)
   (/ x (/ 0.1111111111111111 (* x 0.3333333333333333)))
   (* -3.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x / (0.1111111111111111 / (x * 0.3333333333333333));
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = x / (0.1111111111111111d0 / (x * 0.3333333333333333d0))
    else
        tmp = (-3.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x / (0.1111111111111111 / (x * 0.3333333333333333));
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = x / (0.1111111111111111 / (x * 0.3333333333333333))
	else:
		tmp = -3.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(x / Float64(0.1111111111111111 / Float64(x * 0.3333333333333333)));
	else
		tmp = Float64(-3.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = x / (0.1111111111111111 / (x * 0.3333333333333333));
	else
		tmp = -3.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(x / N[(0.1111111111111111 / N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   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 67.5%

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

   7. Taylor expanded in x around 0 67.5%

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

      1. *-commutative 67.5%

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

   9. Simplified 67.5%

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

       1. clear-num 67.5%

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

       2. un-div-inv 67.6%

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

       3. *-un-lft-identity 67.6%

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

       4. *-commutative 67.6%

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

       5. times-frac 67.6%

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

       6. metadata-eval 67.6%

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

   11. Applied egg-rr 67.6%

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

   12. Taylor expanded in x around 0 85.8%

       \[\leadsto \frac{x}{0.1111111111111111 \cdot \color{blue}{\frac{3}{x}}} \]
   13. Step-by-step derivation

       1. clear-num 85.7%

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

       2. un-div-inv 85.7%

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

       3. div-inv 85.8%

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

       4. metadata-eval 85.8%

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

   14. Applied egg-rr 85.8%

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

   if 1.5 < 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* 99.8%

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

   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 0.3%

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

   7. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   9. Simplified 0.5%

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

       1. clear-num 0.5%

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

       2. un-div-inv 0.5%

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

       3. *-un-lft-identity 0.5%

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

       4. *-commutative 0.5%

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

       5. times-frac 0.5%

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

       6. metadata-eval 0.5%

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

   11. Applied egg-rr 0.5%

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

   12. Taylor expanded in x around 0 0.3%

       \[\leadsto \frac{x}{0.1111111111111111 \cdot \color{blue}{\frac{3}{x}}} \]
   13. Step-by-step derivation

       1. clear-num 0.3%

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

       2. un-div-inv 0.3%

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

       3. div-inv 0.3%

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

       4. metadata-eval 0.3%

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

   14. Applied egg-rr 0.3%

       \[\leadsto \frac{x}{\color{blue}{\frac{0.1111111111111111}{x \cdot 0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. clear-num 0.3%

          \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{x \cdot 0.3333333333333333}{0.1111111111111111}}}} \]

       2. associate-/l* 0.3%

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \frac{0.3333333333333333}{0.1111111111111111}}}} \]

       3. metadata-eval 0.3%

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

       4. *-commutative 0.3%

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

       5. frac-2neg 0.3%

          \[\leadsto \color{blue}{\frac{-x}{-\frac{1}{3 \cdot x}}} \]

       6. associate-/r* 0.3%

          \[\leadsto \frac{-x}{-\color{blue}{\frac{\frac{1}{3}}{x}}} \]

       7. metadata-eval 0.3%

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

       8. distribute-frac-neg2 0.3%

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

       9. *-un-lft-identity 0.3%

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

       10. metadata-eval 0.3%

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

       11. frac-2neg 0.3%

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

       12. div-inv 0.3%

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

       13. times-frac 0.3%

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

       14. metadata-eval 0.3%

           \[\leadsto \color{blue}{-3} \cdot \frac{-x}{\frac{1}{x}} \]

       15. add-sqr-sqrt 0.0%

           \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{1}{x}} \]

       16. sqrt-unprod 53.6%

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

       17. sqr-neg 53.6%

           \[\leadsto -3 \cdot \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{1}{x}} \]

       18. sqrt-unprod 53.6%

           \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{1}{x}} \]

       19. add-sqr-sqrt 53.6%

           \[\leadsto -3 \cdot \frac{\color{blue}{x}}{\frac{1}{x}} \]

   16. Applied egg-rr 53.6%

       \[\leadsto \color{blue}{-3 \cdot \frac{x}{\frac{1}{x}}} \]
   17. Step-by-step derivation

       1. pow1 53.6%

          \[\leadsto -3 \cdot \frac{\color{blue}{{x}^{1}}}{\frac{1}{x}} \]

       2. inv-pow 53.6%

          \[\leadsto -3 \cdot \frac{{x}^{1}}{\color{blue}{{x}^{-1}}} \]

       3. pow-div 53.6%

          \[\leadsto -3 \cdot \color{blue}{{x}^{\left(1 - -1\right)}} \]

       4. metadata-eval 53.6%

          \[\leadsto -3 \cdot {x}^{\color{blue}{2}} \]

       5. pow2 53.6%

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

   18. Applied egg-rr 53.6%

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

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) (* x (* x 3.0)) (* -3.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = x * (x * 3.0d0)
    else
        tmp = (-3.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = x * (x * 3.0)
	else:
		tmp = -3.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = Float64(-3.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = x * (x * 3.0);
	else
		tmp = -3.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   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 85.8%

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

   if 1.5 < 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* 99.8%

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

   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 0.3%

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

   7. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   9. Simplified 0.5%

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

       1. clear-num 0.5%

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

       2. un-div-inv 0.5%

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

       3. *-un-lft-identity 0.5%

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

       4. *-commutative 0.5%

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

       5. times-frac 0.5%

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

       6. metadata-eval 0.5%

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

   11. Applied egg-rr 0.5%

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

   12. Taylor expanded in x around 0 0.3%

       \[\leadsto \frac{x}{0.1111111111111111 \cdot \color{blue}{\frac{3}{x}}} \]
   13. Step-by-step derivation

       1. clear-num 0.3%

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

       2. un-div-inv 0.3%

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

       3. div-inv 0.3%

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

       4. metadata-eval 0.3%

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

   14. Applied egg-rr 0.3%

       \[\leadsto \frac{x}{\color{blue}{\frac{0.1111111111111111}{x \cdot 0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. clear-num 0.3%

          \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{x \cdot 0.3333333333333333}{0.1111111111111111}}}} \]

       2. associate-/l* 0.3%

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \frac{0.3333333333333333}{0.1111111111111111}}}} \]

       3. metadata-eval 0.3%

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

       4. *-commutative 0.3%

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

       5. frac-2neg 0.3%

          \[\leadsto \color{blue}{\frac{-x}{-\frac{1}{3 \cdot x}}} \]

       6. associate-/r* 0.3%

          \[\leadsto \frac{-x}{-\color{blue}{\frac{\frac{1}{3}}{x}}} \]

       7. metadata-eval 0.3%

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

       8. distribute-frac-neg2 0.3%

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

       9. *-un-lft-identity 0.3%

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

       10. metadata-eval 0.3%

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

       11. frac-2neg 0.3%

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

       12. div-inv 0.3%

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

       13. times-frac 0.3%

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

       14. metadata-eval 0.3%

           \[\leadsto \color{blue}{-3} \cdot \frac{-x}{\frac{1}{x}} \]

       15. add-sqr-sqrt 0.0%

           \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{1}{x}} \]

       16. sqrt-unprod 53.6%

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

       17. sqr-neg 53.6%

           \[\leadsto -3 \cdot \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{1}{x}} \]

       18. sqrt-unprod 53.6%

           \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{1}{x}} \]

       19. add-sqr-sqrt 53.6%

           \[\leadsto -3 \cdot \frac{\color{blue}{x}}{\frac{1}{x}} \]

   16. Applied egg-rr 53.6%

       \[\leadsto \color{blue}{-3 \cdot \frac{x}{\frac{1}{x}}} \]
   17. Step-by-step derivation

       1. pow1 53.6%

          \[\leadsto -3 \cdot \frac{\color{blue}{{x}^{1}}}{\frac{1}{x}} \]

       2. inv-pow 53.6%

          \[\leadsto -3 \cdot \frac{{x}^{1}}{\color{blue}{{x}^{-1}}} \]

       3. pow-div 53.6%

          \[\leadsto -3 \cdot \color{blue}{{x}^{\left(1 - -1\right)}} \]

       4. metadata-eval 53.6%

          \[\leadsto -3 \cdot {x}^{\color{blue}{2}} \]

       5. pow2 53.6%

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

   18. Applied egg-rr 53.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-3.0 * N[(x * x), $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. 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 48.8%

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

7. Taylor expanded in x around 0 49.0%

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

   1. *-commutative 49.0%

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

9. Simplified 49.0%

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

    1. clear-num 48.9%

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

    2. un-div-inv 49.0%

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

    3. *-un-lft-identity 49.0%

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

    4. *-commutative 49.0%

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

    5. times-frac 49.0%

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

    6. metadata-eval 49.0%

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

11. Applied egg-rr 49.0%

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

12. Taylor expanded in x around 0 62.1%

    \[\leadsto \frac{x}{0.1111111111111111 \cdot \color{blue}{\frac{3}{x}}} \]
13. Step-by-step derivation

    1. clear-num 62.1%

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

    2. un-div-inv 62.1%

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

    3. div-inv 62.1%

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

    4. metadata-eval 62.1%

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

14. Applied egg-rr 62.1%

    \[\leadsto \frac{x}{\color{blue}{\frac{0.1111111111111111}{x \cdot 0.3333333333333333}}} \]
15. Step-by-step derivation

    1. clear-num 62.1%

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{x \cdot 0.3333333333333333}{0.1111111111111111}}}} \]

    2. associate-/l* 62.1%

       \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \frac{0.3333333333333333}{0.1111111111111111}}}} \]

    3. metadata-eval 62.1%

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

    4. *-commutative 62.1%

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

    5. frac-2neg 62.1%

       \[\leadsto \color{blue}{\frac{-x}{-\frac{1}{3 \cdot x}}} \]

    6. associate-/r* 62.1%

       \[\leadsto \frac{-x}{-\color{blue}{\frac{\frac{1}{3}}{x}}} \]

    7. metadata-eval 62.1%

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

    8. distribute-frac-neg2 62.1%

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

    9. *-un-lft-identity 62.1%

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

    10. metadata-eval 62.1%

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

    11. frac-2neg 62.1%

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

    12. div-inv 62.1%

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

    13. times-frac 62.1%

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

    14. metadata-eval 62.1%

        \[\leadsto \color{blue}{-3} \cdot \frac{-x}{\frac{1}{x}} \]

    15. add-sqr-sqrt 41.4%

        \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{1}{x}} \]

    16. sqrt-unprod 66.4%

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

    17. sqr-neg 66.4%

        \[\leadsto -3 \cdot \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{1}{x}} \]

    18. sqrt-unprod 25.0%

        \[\leadsto -3 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{1}{x}} \]

    19. add-sqr-sqrt 42.4%

        \[\leadsto -3 \cdot \frac{\color{blue}{x}}{\frac{1}{x}} \]

16. Applied egg-rr 42.4%

    \[\leadsto \color{blue}{-3 \cdot \frac{x}{\frac{1}{x}}} \]
17. Step-by-step derivation

    1. pow1 42.4%

       \[\leadsto -3 \cdot \frac{\color{blue}{{x}^{1}}}{\frac{1}{x}} \]

    2. inv-pow 42.4%

       \[\leadsto -3 \cdot \frac{{x}^{1}}{\color{blue}{{x}^{-1}}} \]

    3. pow-div 42.4%

       \[\leadsto -3 \cdot \color{blue}{{x}^{\left(1 - -1\right)}} \]

    4. metadata-eval 42.4%

       \[\leadsto -3 \cdot {x}^{\color{blue}{2}} \]

    5. pow2 42.4%

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

18. Applied egg-rr 42.4%

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

Alternative 5: 3.1% 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 48.8%

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

7. Taylor expanded in x around 0 49.0%

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

   1. *-commutative 49.0%

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

9. Simplified 49.0%

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

10. Taylor expanded in x around inf 3.0%

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

Alternative 6: 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 48.8%

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

7. Taylor expanded in x around 0 49.0%

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

   1. *-commutative 49.0%

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

9. Simplified 49.0%

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

10. Taylor expanded in x around inf 1.6%

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

    1. associate-*r/ 1.6%

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

    2. metadata-eval 1.6%

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

12. Simplified 1.6%

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

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 2024107 
(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))))