ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.3%

Time: 10.0s

Alternatives: 16

Speedup: 1.0×

Specification

\[1.99 \leq x \land x \leq 2.01\]

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{-20}\right)}^{x}\right)}^{\left(x \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp -20.0) x) (* x -0.5))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(-20.0), x), (x * -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp((-20.0d0)) ** x) ** (x * (-0.5d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(-20.0), x), (x * -0.5));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(-20.0), x), (x * -0.5))

Julia

function code(x)
	return Float64(cos(x) * ((exp(-20.0) ^ x) ^ Float64(x * -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(-20.0) ^ x) ^ (x * -0.5));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[-20.0], $MachinePrecision], x], $MachinePrecision], N[(x * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   4. sqr-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)} \cdot {\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}} \]

   6. frac-2neg N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x \cdot x\right)}{\mathsf{neg}\left(2\right)}\right)}} \]

   7. div-inv N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   8. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(-x\right)\right)}\right)}^{-0.5}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}^{\frac{-1}{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}}^{\frac{-1}{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}\right)}^{\frac{-1}{2}} \]

   4. pow-unpow N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}\right)}}^{\frac{-1}{2}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \cos x \cdot {\left({\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)}^{\frac{-1}{2}} \]

   6. neg-mul-1 N/A

      \[\leadsto \cos x \cdot {\left({\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(-1 \cdot x\right)}}\right)}^{\frac{-1}{2}} \]

   7. pow-unpow N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left({\left(e^{20}\right)}^{x}\right)}^{-1}\right)}^{x}\right)}}^{\frac{-1}{2}} \]

   8. pow-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left({\left(e^{20}\right)}^{x}\right)}^{-1}\right)}^{\left(x \cdot \frac{-1}{2}\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left({\left(e^{20}\right)}^{x}\right)}^{-1}\right)}^{\left(x \cdot \frac{-1}{2}\right)}} \]

6. Applied rewrites 94.8%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-20 \cdot x}\right)}^{\left(x \cdot -0.5\right)}} \]
7. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{-20 \cdot x}\right)}}^{\left(x \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-20 \cdot x}}\right)}^{\left(x \cdot \frac{-1}{2}\right)} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{-20}\right)}^{x}\right)}}^{\left(x \cdot \frac{-1}{2}\right)} \]

   4. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{-20}\right)}^{x}\right)}}^{\left(x \cdot \frac{-1}{2}\right)} \]

   5. lower-exp.f64 99.3

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{-20}\right)}}^{x}\right)}^{\left(x \cdot -0.5\right)} \]

8. Applied rewrites 99.3%

   \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{-20}\right)}^{x}\right)}}^{\left(x \cdot -0.5\right)} \]
9. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{-10}\right)}^{x}\right)}^{\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp -10.0) x) (- x))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(-10.0), x), -x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp((-10.0d0)) ** x) ** -x)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(-10.0), x), -x);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(-10.0), x), -x)

Julia

function code(x)
	return Float64(cos(x) * ((exp(-10.0) ^ x) ^ Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(-10.0) ^ x) ^ -x);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[-10.0], $MachinePrecision], x], $MachinePrecision], (-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   4. sqr-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)} \cdot {\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}} \]

   6. frac-2neg N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x \cdot x\right)}{\mathsf{neg}\left(2\right)}\right)}} \]

   7. div-inv N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   8. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(-x\right)\right)}\right)}^{-0.5}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}^{\frac{-1}{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}}^{\frac{-1}{2}} \]

   3. pow-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{-1}{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot {\left(e^{20}\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot \left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}} \]

   5. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   7. pow-to-exp N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left(e^{20}\right) \cdot \frac{-1}{2}}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   8. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{\log \color{blue}{\left(e^{20}\right)} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   9. rem-log-exp N/A

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{20} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\frac{-1}{2} \cdot 20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   12. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{\log \left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   14. lower-exp.f64 N/A

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\frac{-1}{2} \cdot \log \left(e^{20}\right)}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   15. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   16. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   17. metadata-eval 95.5

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(-x\right)\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   19. lift-neg.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)} \]

   20. distribute-rgt-neg-out N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   21. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)} \]

   22. lower-neg.f64 95.5

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(-x \cdot x\right)}} \]

6. Applied rewrites 95.5%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-10}\right)}^{\left(-x \cdot x\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)} \]

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

      \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   5. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{-10}\right)}^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{-10}\right)}^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{-10}\right)}^{x}\right)}}^{\left(\mathsf{neg}\left(x\right)\right)} \]

   8. lower-neg.f64 98.1

      \[\leadsto \cos x \cdot {\left({\left(e^{-10}\right)}^{x}\right)}^{\color{blue}{\left(-x\right)}} \]

8. Applied rewrites 98.1%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{-10}\right)}^{x}\right)}^{\left(-x\right)}} \]
9. Add Preprocessing

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 10.0) x) x)))

C

double code(double x) {
	return cos(x) * pow(pow(exp(10.0), x), x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(10.0d0) ** x) ** x)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(10.0), x), x);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(10.0), x), x)

Julia

function code(x)
	return Float64(cos(x) * ((exp(10.0) ^ x) ^ x))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(10.0) ^ x) ^ x);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[10.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   4. sqr-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)} \cdot {\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}} \]

   6. frac-2neg N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x \cdot x\right)}{\mathsf{neg}\left(2\right)}\right)}} \]

   7. div-inv N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   8. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(-x\right)\right)}\right)}^{-0.5}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}^{\frac{-1}{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}}^{\frac{-1}{2}} \]

   3. pow-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{-1}{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot {\left(e^{20}\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot \left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}} \]

   5. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   7. pow-to-exp N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left(e^{20}\right) \cdot \frac{-1}{2}}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   8. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{\log \color{blue}{\left(e^{20}\right)} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   9. rem-log-exp N/A

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{20} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\frac{-1}{2} \cdot 20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   12. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{\log \left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   14. lower-exp.f64 N/A

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\frac{-1}{2} \cdot \log \left(e^{20}\right)}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   15. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   16. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   17. metadata-eval 95.5

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(-x\right)\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   19. lift-neg.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)} \]

   20. distribute-rgt-neg-out N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   21. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)} \]

   22. lower-neg.f64 95.5

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(-x \cdot x\right)}} \]

6. Applied rewrites 95.5%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-10}\right)}^{\left(-x \cdot x\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   2. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{-10}\right)}}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

   3. pow-exp N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{-10 \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \cos x \cdot e^{-10 \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   5. neg-mul-1 N/A

      \[\leadsto \cos x \cdot e^{-10 \cdot \color{blue}{\left(-1 \cdot \left(x \cdot x\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(-10 \cdot -1\right) \cdot \left(x \cdot x\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10} \cdot \left(x \cdot x\right)} \]

   8. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\color{blue}{\left(x \cdot x\right)}} \]

   10. pow-unpow N/A

       \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]

   12. lower-pow.f64 N/A

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

   13. lower-exp.f64 97.9

       \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{10}\right)}}^{x}\right)}^{x} \]

8. Applied rewrites 97.9%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]
9. Add Preprocessing

Alternative 4: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{-10}\right)}^{\left(x \cdot \left(-x\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp -10.0) (* x (- x)))))

C

double code(double x) {
	return cos(x) * pow(exp(-10.0), (x * -x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((-10.0d0)) ** (x * -x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(-10.0), (x * -x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(-10.0), (x * -x))

Julia

function code(x)
	return Float64(cos(x) * (exp(-10.0) ^ Float64(x * Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(-10.0) ^ (x * -x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[-10.0], $MachinePrecision], N[(x * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   4. sqr-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)} \cdot {\left(e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{\left(\frac{x \cdot x}{2}\right)}} \]

   6. frac-2neg N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x \cdot x\right)}{\mathsf{neg}\left(2\right)}\right)}} \]

   7. div-inv N/A

      \[\leadsto \cos x \cdot {\left(e^{10} \cdot e^{10}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   8. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(-x\right)\right)}\right)}^{-0.5}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}^{\frac{-1}{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}}^{\frac{-1}{2}} \]

   3. pow-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{-1}{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot {\left(e^{20}\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot \left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}} \]

   5. pow-unpow N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   7. pow-to-exp N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left(e^{20}\right) \cdot \frac{-1}{2}}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   8. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\left(e^{\log \color{blue}{\left(e^{20}\right)} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   9. rem-log-exp N/A

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{20} \cdot \frac{-1}{2}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\frac{-1}{2} \cdot 20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   12. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{\log \left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   14. lower-exp.f64 N/A

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\frac{-1}{2} \cdot \log \left(e^{20}\right)}\right)}}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   15. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \log \color{blue}{\left(e^{20}\right)}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   16. rem-log-exp N/A

       \[\leadsto \cos x \cdot {\left(e^{\frac{-1}{2} \cdot \color{blue}{20}}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   17. metadata-eval 95.5

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{-10}}\right)}^{\left(x \cdot \left(-x\right)\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

   19. lift-neg.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)} \]

   20. distribute-rgt-neg-out N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   21. lift-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(\mathsf{neg}\left(\color{blue}{x \cdot x}\right)\right)} \]

   22. lower-neg.f64 95.5

       \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\color{blue}{\left(-x \cdot x\right)}} \]

6. Applied rewrites 95.5%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{-10}\right)}^{\left(-x \cdot x\right)}} \]

7. Final simplification 95.5%

   \[\leadsto \cos x \cdot {\left(e^{-10}\right)}^{\left(x \cdot \left(-x\right)\right)} \]
8. Add Preprocessing

Alternative 5: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{x \cdot x}\right)}^{10} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* x x)) 10.0)))

C

double code(double x) {
	return cos(x) * pow(exp((x * x)), 10.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x * x)) ** 10.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp((x * x)), 10.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp((x * x)), 10.0)

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(x * x)) ^ 10.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x * x)) ^ 10.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision], 10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp 10.0) (* x x))))

C

double code(double x) {
	return cos(x) * pow(exp(10.0), (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(10.0d0) ** (x * x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(10.0), (x * x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(10.0), (x * x))

Julia

function code(x)
	return Float64(cos(x) * (exp(10.0) ^ Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(10.0) ^ (x * x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[10.0], $MachinePrecision], N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   3. exp-prod N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   5. lower-exp.f64 95.3

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{10}\right)}}^{\left(x \cdot x\right)} \]

4. Applied rewrites 95.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 27.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot e^{x \cdot \left(x \cdot 10\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) -0.001388888888888889 0.041666666666666664) -0.5)
   1.0)
  (exp (* x (* x 10.0)))))

C

double code(double x) {
	return fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0) * exp((x * (x * 10.0)));
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0) * exp(Float64(x * Float64(x * 10.0))))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos x \cdot e^{10 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   4. lower-*.f64 N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(x \cdot 10\right)} \cdot x} \]

   6. lower-*.f64 94.3

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(x \cdot 10\right)} \cdot x} \]

4. Applied rewrites 94.3%

   \[\leadsto \cos x \cdot e^{\color{blue}{\left(x \cdot 10\right) \cdot x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot e^{\left(x \cdot 10\right) \cdot x} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

   14. lower-*.f64 27.5

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot e^{\left(x \cdot 10\right) \cdot x} \]

7. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot e^{\left(x \cdot 10\right) \cdot x} \]

8. Final simplification 27.5%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot e^{x \cdot \left(x \cdot 10\right)} \]
9. Add Preprocessing

Alternative 9: 27.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* 10.0 (* x x)))
  (fma
   x
   (*
    x
    (fma
     (* x x)
     (fma (* x x) -0.001388888888888889 0.041666666666666664)
     -0.5))
   1.0)))

C

double code(double x) {
	return exp((10.0 * (x * x))) * fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
}

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0))
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   10. lower-*.f64 21.3

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Applied rewrites 21.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 21.3%

   \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \color{blue}{0.041666666666666664}, \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right)} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

10. Applied rewrites 27.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

11. Final simplification 27.5%

    \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]
12. Add Preprocessing

Alternative 10: 21.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* 10.0 (* x x)))
  (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0)))

C

double code(double x) {
	return exp((10.0 * (x * x))) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
}

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0))
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   10. lower-*.f64 21.3

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Applied rewrites 21.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{24} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{1}{24} \cdot x\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   13. lower-*.f64 21.3

       \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

8. Applied rewrites 21.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

9. Final simplification 21.3%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) \]
10. Add Preprocessing

Alternative 11: 18.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* 10.0 (* x x))) (fma x (* x -0.5) 1.0)))

C

double code(double x) {
	return exp((10.0 * (x * x))) * fma(x, (x * -0.5), 1.0);
}

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * fma(x, Float64(x * -0.5), 1.0))
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. lower-*.f64 18.2

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Applied rewrites 18.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

6. Final simplification 18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \]
7. Add Preprocessing

Alternative 12: 10.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 166.66666666666666, 50\right), 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma x (* x -0.5) 1.0)
  (fma (* x x) (fma x (* x (fma (* x x) 166.66666666666666 50.0)) 10.0) 1.0)))

C

double code(double x) {
	return fma(x, (x * -0.5), 1.0) * fma((x * x), fma(x, (x * fma((x * x), 166.66666666666666, 50.0)), 10.0), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 166.66666666666666, 50.0)), 10.0), 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 166.66666666666666 + 50.0), $MachinePrecision]), $MachinePrecision] + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]

5. Taylor expanded in x around 0

   \[\leadsto \cos x \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 9.6%

   \[\leadsto \cos x \cdot \color{blue}{1} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 9.7

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot 1 \]

10. Applied rewrites 9.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot 1 \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right)} \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right) + 10}, 1\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right) + 10, 1\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right)} + 10, 1\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 10\right)}, 1\right) \]

    9. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)}, 10\right), 1\right) \]

    10. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{500}{3} \cdot {x}^{2} + 50\right)}, 10\right), 1\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{500}{3}} + 50\right), 10\right), 1\right) \]

    12. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{500}{3}, 50\right)}, 10\right), 1\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{500}{3}, 50\right), 10\right), 1\right) \]

    14. lower-*.f64 10.3

        \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 166.66666666666666, 50\right), 10\right), 1\right) \]

13. Applied rewrites 10.3%

    \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 166.66666666666666, 50\right), 10\right), 1\right)} \]
14. Add Preprocessing

Alternative 13: 10.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 50, 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma x (* x -0.5) 1.0) (fma x (* x (fma x (* x 50.0) 10.0)) 1.0)))

C

double code(double x) {
	return fma(x, (x * -0.5), 1.0) * fma(x, (x * fma(x, (x * 50.0), 10.0)), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * -0.5), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 50.0), 10.0)), 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 50.0), $MachinePrecision] + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]

5. Taylor expanded in x around 0

   \[\leadsto \cos x \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 9.6%

   \[\leadsto \cos x \cdot \color{blue}{1} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 9.7

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot 1 \]

10. Applied rewrites 9.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot 1 \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right) + 1\right)} \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(10 + 50 \cdot {x}^{2}\right) + 1\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(10 + 50 \cdot {x}^{2}\right)\right)} + 1\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(10 + 50 \cdot {x}^{2}\right), 1\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(10 + 50 \cdot {x}^{2}\right)}, 1\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(50 \cdot {x}^{2} + 10\right)}, 1\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot 50} + 10\right), 1\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 50 + 10\right), 1\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot 50\right)} + 10\right), 1\right) \]

    10. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 50, 10\right)}, 1\right) \]

    11. lower-*.f64 10.1

        \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 50}, 10\right), 1\right) \]

13. Applied rewrites 10.1%

    \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 50, 10\right), 1\right)} \]
14. Add Preprocessing

Alternative 14: 9.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(10, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma x (* x -0.5) 1.0) (fma 10.0 (* x x) 1.0)))

C

double code(double x) {
	return fma(x, (x * -0.5), 1.0) * fma(10.0, (x * x), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * -0.5), 1.0) * fma(10.0, Float64(x * x), 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(10.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]

5. Taylor expanded in x around 0

   \[\leadsto \cos x \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 9.6%

   \[\leadsto \cos x \cdot \color{blue}{1} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 9.7

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot 1 \]

10. Applied rewrites 9.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot 1 \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + 10 \cdot {x}^{2}\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(10 \cdot {x}^{2} + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(10, {x}^{2}, 1\right)} \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(10, \color{blue}{x \cdot x}, 1\right) \]

    4. lower-*.f64 9.9

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(10, \color{blue}{x \cdot x}, 1\right) \]

13. Applied rewrites 9.9%

    \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(10, x \cdot x, 1\right)} \]
14. Add Preprocessing

Alternative 15: 9.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(x \cdot -0.5\right)\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x (* x -0.5)) 1.0))

C

double code(double x) {
	return (x * (x * -0.5)) * 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * (-0.5d0))) * 1.0d0
end function

Java

public static double code(double x) {
	return (x * (x * -0.5)) * 1.0;
}

Python

def code(x):
	return (x * (x * -0.5)) * 1.0

Julia

function code(x)
	return Float64(Float64(x * Float64(x * -0.5)) * 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * -0.5)) * 1.0;
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]

5. Taylor expanded in x around 0

   \[\leadsto \cos x \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 9.6%

   \[\leadsto \cos x \cdot \color{blue}{1} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 9.7

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot 1 \]

10. Applied rewrites 9.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot 1 \]

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.7%

    \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot -0.5\right)}\right) \cdot 1 \]
14. Add Preprocessing

Alternative 16: 1.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 1"
  :precision binary64
  :pre (and (<= 1.99 x) (<= x 2.01))
  (* (cos x) (exp (* 10.0 (* x x)))))