ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.3%

Time: 10.0s

Alternatives: 17

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 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   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. sqr-pow N/A

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

   5. sqr-pow N/A

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

   6. lift-*.f64 N/A

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

   7. pow-unpow N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. pow-unpow N/A

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

   11. pow-pow N/A

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

   12. 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 94.7%

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

   1. exp-prod N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-exp.f64 99.2

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

8. Applied rewrites 99.2%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{x + x}\right)}^{x}\right)}^{5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp (+ x x)) x) 5.0)))

C

double code(double x) {
	return cos(x) * pow(pow(exp((x + x)), x), 5.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp((x + x)) ** x) ** 5.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp((x + x)), x), 5.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp((x + x)), x), 5.0)

Julia

function code(x)
	return Float64(cos(x) * ((exp(Float64(x + x)) ^ x) ^ 5.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp((x + x)) ^ x) ^ 5.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision], x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

6. Applied rewrites 95.1%

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

8. Applied rewrites 97.6%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{x}\right)}^{\left(x + x\right)}\right)}^{5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp x) (+ x x)) 5.0)))

C

double code(double x) {
	return cos(x) * pow(pow(exp(x), (x + x)), 5.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(x) ** (x + x)) ** 5.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(x), (x + x)), 5.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(x), (x + x)), 5.0)

Julia

function code(x)
	return Float64(cos(x) * ((exp(x) ^ Float64(x + x)) ^ 5.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(x) ^ (x + x)) ^ 5.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[x], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

6. Applied rewrites 95.1%

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

8. Applied rewrites 96.7%

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

Alternative 4: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{x \cdot \left(x + x\right)}\right)}^{5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* x (+ x x))) 5.0)))

C

double code(double x) {
	return cos(x) * pow(exp((x * (x + x))), 5.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x * (x + x))) ** 5.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp((x * (x + x))), 5.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp((x * (x + x))), 5.0)

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(x * Float64(x + x))) ^ 5.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x * (x + x))) ^ 5.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

6. Applied rewrites 95.1%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{\left(x + x\right) \cdot x}\right)}^{5}} \]

7. Final simplification 95.1%

   \[\leadsto \cos x \cdot {\left(e^{x \cdot \left(x + x\right)}\right)}^{5} \]
8. Add Preprocessing

Alternative 5: 95.3% 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 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   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. sqr-pow N/A

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

   5. sqr-pow N/A

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

   6. pow-pow N/A

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

   7. *-commutative N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. pow-to-exp N/A

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

   11. lift-exp.f64 N/A

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

   12. rem-log-exp N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-exp.f64 N/A

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

   16. metadata-eval 95.1

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

   17. lift-*.f64 N/A

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

   18. lift-neg.f64 N/A

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

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

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

   20. lift-*.f64 N/A

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

   21. lower-neg.f64 95.1

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

6. Applied rewrites 95.1%

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

7. Final simplification 95.1%

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

Alternative 6: 95.3% 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 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 95.1

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

4. Applied rewrites 95.1%

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

Alternative 7: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{1}{e^{-10 \cdot \left(x \cdot x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (/ 1.0 (exp (* -10.0 (* x x))))))

C

double code(double x) {
	return cos(x) * (1.0 / exp((-10.0 * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (1.0d0 / exp(((-10.0d0) * (x * x))))
end function

Java

public static double code(double x) {
	return Math.cos(x) * (1.0 / Math.exp((-10.0 * (x * x))));
}

Python

def code(x):
	return math.cos(x) * (1.0 / math.exp((-10.0 * (x * x))))

Julia

function code(x)
	return Float64(cos(x) * Float64(1.0 / exp(Float64(-10.0 * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (1.0 / exp((-10.0 * (x * x))));
end

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. pow-unpow N/A

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

   8. 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 95.1%

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

6. Applied rewrites 94.1%

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

7. Final simplification 94.1%

   \[\leadsto \cos x \cdot \frac{1}{e^{-10 \cdot \left(x \cdot x\right)}} \]
8. Add Preprocessing

Alternative 8: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(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[Exp[N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

3. Final simplification 94.1%

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

Alternative 9: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	return exp((x * (x * 10.0))) * 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(x * Float64(x * 10.0))) * 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[(x * N[(x * 10.0), $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 94.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\cos x \cdot e^{10 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{e^{10 \cdot {x}^{2}} \cdot \cos x} \]

   2. *-lft-identity N/A

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

   3. lower-*.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. lower-cos.f64 94.0

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

5. Applied rewrites 94.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 27.5

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

8. Applied rewrites 27.5%

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

Alternative 10: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. Final simplification 21.3%

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

Alternative 11: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\cos x \cdot e^{10 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{e^{10 \cdot {x}^{2}} \cdot \cos x} \]

   2. *-lft-identity N/A

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

   3. lower-*.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. lower-cos.f64 94.0

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

5. Applied rewrites 94.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 21.3

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

8. Applied rewrites 21.3%

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

Alternative 12: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\cos x \cdot e^{10 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{e^{10 \cdot {x}^{2}} \cdot \cos x} \]

   2. *-lft-identity N/A

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

   3. lower-*.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. lower-cos.f64 94.0

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

5. Applied rewrites 94.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 18.2

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

8. Applied rewrites 18.2%

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

Alternative 13: 10.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 166.66666666666666, 50\right), 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (* x x) (fma (* x x) 0.041666666666666664 -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), fma((x * x), 0.041666666666666664, -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(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0) * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 166.66666666666666), 50.0), 10.0), 1.0))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 166.66666666666666), $MachinePrecision] + 50.0), $MachinePrecision] + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right)\right)} \]

8. Applied rewrites 10.2%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 166.66666666666666, 50\right), 10\right), 1\right)} \]
9. Add Preprocessing

Alternative 14: 10.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 50, 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (* x x) (fma (* x x) 0.041666666666666664 -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), fma((x * x), 0.041666666666666664, -0.5), 1.0) * fma((x * x), fma((x * x), 50.0, 10.0), 1.0);
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0) * fma(Float64(x * x), fma(Float64(x * x), 50.0, 10.0), 1.0))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 50.0 + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right) + 1\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 10 + 50 \cdot {x}^{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + 50 \cdot {x}^{2}, 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + 50 \cdot {x}^{2}, 1\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{50 \cdot {x}^{2} + 10}, 1\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot 50} + 10, 1\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, 50, 10\right)}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 50, 10\right), 1\right) \]

   9. lower-*.f64 10.0

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 50, 10\right), 1\right) \]

8. Applied rewrites 10.0%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 50, 10\right), 1\right)} \]
9. Add Preprocessing

Alternative 15: 9.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 9.8

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

8. Applied rewrites 9.8%

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

Alternative 16: 9.7% accurate, 19.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 9.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 9.7

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

8. Applied rewrites 9.7%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 9.7

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

11. Applied rewrites 9.7%

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

Alternative 17: 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 94.1%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 2024214 
(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)))))