ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.0%

Time: 13.7s

Alternatives: 13

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \left({\left({\left({\left(e^{5}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (cos x)
  (*
   (pow (pow (pow (exp 5.0) 2.0) x) (/ x 2.0))
   (pow (pow (exp 10.0) x) (/ x 2.0)))))

C

double code(double x) {
	return cos(x) * (pow(pow(pow(exp(5.0), 2.0), x), (x / 2.0)) * pow(pow(exp(10.0), x), (x / 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((((exp(5.0d0) ** 2.0d0) ** x) ** (x / 2.0d0)) * ((exp(10.0d0) ** x) ** (x / 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * (Math.pow(Math.pow(Math.pow(Math.exp(5.0), 2.0), x), (x / 2.0)) * Math.pow(Math.pow(Math.exp(10.0), x), (x / 2.0)));
}

Python

def code(x):
	return math.cos(x) * (math.pow(math.pow(math.pow(math.exp(5.0), 2.0), x), (x / 2.0)) * math.pow(math.pow(math.exp(10.0), x), (x / 2.0)))

Julia

function code(x)
	return Float64(cos(x) * Float64((((exp(5.0) ^ 2.0) ^ x) ^ Float64(x / 2.0)) * ((exp(10.0) ^ x) ^ Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((((exp(5.0) ^ 2.0) ^ x) ^ (x / 2.0)) * ((exp(10.0) ^ x) ^ (x / 2.0)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Power[N[Power[N[Power[N[Exp[5.0], $MachinePrecision], 2.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Exp[10.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. pow-prod-up N/A

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

   8. sqr-pow N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 98.0%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 99.0

      \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{10}\right)}^{0.5} \cdot \color{blue}{{\left(e^{10}\right)}^{0.5}}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)}\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \cos x \cdot \left({\left({\color{blue}{\left({\left(e^{10}\right)}^{0.5} \cdot {\left(e^{10}\right)}^{0.5}\right)}}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.0

      \[\leadsto \cos x \cdot \left({\left({\color{blue}{\left({\left({\left(e^{10}\right)}^{0.5}\right)}^{2}\right)}}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)}\right) \]

   4. lift-pow.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. pow-exp N/A

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

   7. lower-exp.f64 N/A

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

   8. metadata-eval 99.0

      \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{\color{blue}{5}}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)}\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \cos x \cdot \left({\left({\color{blue}{\left({\left(e^{5}\right)}^{2}\right)}}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{5}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot 1}}{2}\right)}\right) \]

   2. *-rgt-identity 99.0

      \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{5}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x}}{2}\right)}\right) \]

10. Applied rewrites 99.0%

    \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{5}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x \cdot 1}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x}}{2}\right)}\right) \]

11. Final simplification 99.0%

    \[\leadsto \cos x \cdot \left({\left({\left({\left(e^{5}\right)}^{2}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}\right) \]
12. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}\\ \cos x \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (pow (exp 10.0) x) (/ x 2.0)))) (* (cos x) (* t_0 t_0))))

C

double code(double x) {
	double t_0 = pow(pow(exp(10.0), x), (x / 2.0));
	return cos(x) * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (exp(10.0d0) ** x) ** (x / 2.0d0)
    code = cos(x) * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.pow(Math.exp(10.0), x), (x / 2.0));
	return Math.cos(x) * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.pow(math.pow(math.exp(10.0), x), (x / 2.0))
	return math.cos(x) * (t_0 * t_0)

Julia

function code(x)
	t_0 = (exp(10.0) ^ x) ^ Float64(x / 2.0)
	return Float64(cos(x) * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = (exp(10.0) ^ x) ^ (x / 2.0);
	tmp = cos(x) * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[N[Exp[10.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. pow-prod-up N/A

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

   8. sqr-pow N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 98.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 98.0

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

6. Applied rewrites 98.0%

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

7. Final simplification 98.0%

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

Alternative 3: 95.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\color{blue}{0}}{\frac{x \cdot x}{2} - \frac{x \cdot x}{2}}\right)} \]

   8. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{0}}\right)} \]

   9. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{2 \cdot 0}}\right)} \]

   10. associate-/l/ N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\color{blue}{\left(\frac{\frac{0}{0}}{2}\right)}} \]

   11. +-inverses N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\frac{\color{blue}{\frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2} - \frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2}}}{0}}{2}\right)} \]

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.4%

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

5. Final simplification 95.4%

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

Alternative 4: 98.0% 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 5: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.8

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

5. Applied rewrites 96.8%

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

6. Final simplification 96.8%

   \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x} \]
7. 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 95.4

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

4. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]
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 94.6%

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

Alternative 8: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 27.6

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

5. Applied rewrites 27.6%

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

Alternative 9: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 21.3

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

5. Applied rewrites 21.3%

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

Alternative 10: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

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

Alternative 11: 9.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (cos x) (fma 10.0 (* x x) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 9.8

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

5. Applied rewrites 9.8%

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

6. Final simplification 9.8%

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

Alternative 12: 9.7% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. pow-prod-up N/A

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

   8. sqr-pow N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 98.0%

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

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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

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%

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

Alternative 13: 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 2024324 
(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)))))