ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.1s

Alternatives: 18

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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\cos x}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (cos x) (pow (pow (exp 20.0) x) (/ x -2.0))))

C

double code(double x) {
	return cos(x) / pow(pow(exp(20.0), x), (x / -2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) / ((exp(20.0d0) ** x) ** (x / (-2.0d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) / Math.pow(Math.pow(Math.exp(20.0), x), (x / -2.0));
}

Python

def code(x):
	return math.cos(x) / math.pow(math.pow(math.exp(20.0), x), (x / -2.0))

Julia

function code(x)
	return Float64(cos(x) / ((exp(20.0) ^ x) ^ Float64(x / -2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) / ((exp(20.0) ^ x) ^ (x / -2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] / N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-in N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow-exp N/A

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

   11. lift-exp.f64 N/A

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

   12. sqr-pow N/A

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

   13. pow-prod-down N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. pow-unpow N/A

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

   19. lower-pow.f64 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{1 \cdot \cos x}{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.5

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\cos x}}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}} \]
9. Add Preprocessing

Alternative 2: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(-\left(\frac{\mathsf{PI}\left(\right)}{2} + x\right)\right) + \mathsf{PI}\left(\right)\right) \cdot {\left({\left(e^{10}\right)}^{\left(\left(-x\right) \cdot x\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sin (+ (- (+ (/ (PI) 2.0) x)) (PI)))
  (pow (pow (exp 10.0) (* (- x) x)) -1.0)))

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

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

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 95.3

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

4. Applied rewrites 95.3%

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

   1. remove-double-neg N/A

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

   2. lift-cos.f64 N/A

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

   3. sin-+PI/2-rev N/A

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

   4. sin-neg-rev N/A

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

   5. sin-+PI-rev N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. lower-PI.f64 95.4

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

6. Applied rewrites 95.4%

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

7. Final simplification 95.4%

   \[\leadsto \sin \left(\left(-\left(\frac{\mathsf{PI}\left(\right)}{2} + x\right)\right) + \mathsf{PI}\left(\right)\right) \cdot {\left({\left(e^{10}\right)}^{\left(\left(-x\right) \cdot x\right)}\right)}^{-1} \]
8. Add Preprocessing

Alternative 3: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\frac{\mathsf{PI}\left(\right)}{2} - x\right) \cdot {\left({\left(e^{10}\right)}^{\left(\left(-x\right) \cdot x\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sin (- (/ (PI) 2.0) x)) (pow (pow (exp 10.0) (* (- x) x)) -1.0)))

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

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

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 95.3

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

4. Applied rewrites 95.3%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-neg.f64 N/A

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

   4. sin-+PI/2-rev N/A

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

   5. lower-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 95.4

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

6. Applied rewrites 95.4%

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

7. Final simplification 95.4%

   \[\leadsto \sin \left(\frac{\mathsf{PI}\left(\right)}{2} - x\right) \cdot {\left({\left(e^{10}\right)}^{\left(\left(-x\right) \cdot x\right)}\right)}^{-1} \]
8. Add Preprocessing

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 20.0) x) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), x), (x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** x) ** (x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), x), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), x), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ x) ^ Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ x) ^ (x / 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-pow N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg-rev N/A

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

   10. pow-unpow N/A

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

   11. sqr-pow N/A

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

   12. pow-prod-down N/A

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

   13. unpow-prod-down N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-exp.f64 N/A

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

   19. lift-exp.f64 N/A

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

   20. exp-lft-sqr-rev N/A

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

   21. lower-exp.f64 N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}} \]
7. Add Preprocessing

Alternative 5: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. lift-neg.f64 N/A

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

   5. lower-pow.f64 98.1

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

   6. lift-pow.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. pow-exp N/A

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

   9. lift-neg.f64 N/A

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

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

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. exp-prod N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-exp.f64 96.8

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

6. Applied rewrites 96.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. pow-exp N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 98.2

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

8. Applied rewrites 98.2%

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

Alternative 6: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. 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 97.9

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

4. Applied rewrites 97.9%

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

Alternative 7: 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 94.5%

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

Alternative 8: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(\mathsf{fma}\left(-1.5, \mathsf{PI}\left(\right), x\right)\right)}{{\left(e^{-10}\right)}^{\left(x \cdot x\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (sin (fma -1.5 (PI) x)) (pow (exp -10.0) (* x x))))

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-in N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 94.5%

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

   1. remove-double-neg N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-neg.f64 N/A

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

   9. sin-+PI/2-rev N/A

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

   10. sin-+PI-rev N/A

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

   11. sin-neg-rev N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-PI.f64 N/A

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

   19. lower-PI.f64 94.4

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

6. Applied rewrites 94.4%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{fma}\left(-1.5, \mathsf{PI}\left(\right), x\right)\right)}{{\left(e^{-10}\right)}^{\left(x \cdot x\right)}}} \]
10. Add Preprocessing

Alternative 9: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 10: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{-10 \cdot \left(x \cdot x\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (sin (+ (+ (+ (PI) x) (PI)) (/ (PI) 2.0))) (exp (* -10.0 (* x x)))))

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-in N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 94.5%

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

   1. remove-double-neg N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-+PI-rev N/A

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

   8. cos-+PI-rev N/A

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

   9. sin-+PI/2-rev N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-PI.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-PI.f64 94.6

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

6. Applied rewrites 94.6%

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

7. Final simplification 94.6%

   \[\leadsto \frac{\sin \left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{-10 \cdot \left(x \cdot x\right)}} \]
8. Add Preprocessing

Alternative 11: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sin (+ (- x) (/ (PI) 2.0))) (exp (* 10.0 (* x x)))))

Derivation

1. Initial program 94.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 94.5

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

4. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
5. Add Preprocessing

Alternative 12: 94.3% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-in N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 94.5%

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

   1. remove-double-neg N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-+PI-rev N/A

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

   8. cos-+PI-rev N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-PI.f64 94.6

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

6. Applied rewrites 94.6%

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

7. Final simplification 94.6%

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

Alternative 13: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-in N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow-exp N/A

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

   11. lift-exp.f64 N/A

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

   12. sqr-pow N/A

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

   13. pow-prod-down N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. pow-unpow N/A

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

   19. lower-pow.f64 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{1 \cdot \cos x}{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.5

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\cos x}}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-frac-neg N/A

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

   6. pow-neg N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-unpow N/A

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

   9. lift-exp.f64 N/A

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

   10. associate-/l* N/A

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

   11. pow-exp N/A

       \[\leadsto \frac{\cos x}{\frac{1}{\color{blue}{e^{20 \cdot \frac{x \cdot x}{2}}}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos x}{\frac{1}{e^{\color{blue}{\frac{x \cdot x}{2} \cdot 20}}}} \]

   13. exp-prod N/A

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

   14. associate-/l* N/A

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

   15. pow-exp N/A

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

   16. lift-exp.f64 N/A

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

   17. metadata-eval N/A

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

   18. pow-pow N/A

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

   19. pow2 N/A

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

   20. sqr-pow N/A

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

   21. lift-exp.f64 N/A

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

   22. pow-exp N/A

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

   23. lift-*.f64 N/A

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

   24. pow-exp N/A

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

   25. *-commutative N/A

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

10. Applied rewrites 94.5%

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

Alternative 14: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

Alternative 15: 21.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(fma(Float64(Float64(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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 94.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-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. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 21.3

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

5. Applied rewrites 21.3%

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

Alternative 16: 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 94.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. 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 17: 9.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 95.3

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. exp-lft-sqr-rev N/A

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

   7. rem-log-exp N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 9.8

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

7. Applied rewrites 9.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 9.9

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

10. Applied rewrites 9.9%

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

Alternative 18: 1.5% accurate, 216.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.5%

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

Reproduce

herbie shell --seed 2024338 
(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)))))