ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 98.2%

Time: 4.7s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 98.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (pow (exp 10.0) x) x) (sin (fma (PI) 0.5 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-cos.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. exp-prod N/A

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

   10. pow2 N/A

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

   11. pow-unpow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-+.f64 98.2

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 98.2%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-exp.f64 N/A

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

   5. exp-prod N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. pow-pow 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-pow.f64 N/A

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

   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. lift-neg.f64 95.3

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

6. Applied rewrites 95.3%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 98.1%

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

Alternative 3: 98.0% accurate, 0.5× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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) ^ Float64(-x)) ^ Float64(-x)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-exp.f64 N/A

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

   5. exp-prod N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

Alternative 4: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[Power[N[Exp[10.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision] * N[Cos[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-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. exp-prod N/A

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

   10. pow2 N/A

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

   11. pow-unpow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 5: 95.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (exp 10.0) (* x x)) (sin (fma (PI) 0.5 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-cos.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. exp-prod N/A

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

   10. pow2 N/A

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

   11. pow-unpow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-+.f64 98.2

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 98.2%

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

    1. lift-pow.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. pow-pow N/A

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

    4. lift-*.f64 N/A

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

    5. lift-pow.f64 95.3

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

11. Applied rewrites 95.3%

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

Alternative 6: 95.2% accurate, 0.7× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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(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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-exp.f64 N/A

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

   5. exp-prod N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. pow-pow 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-pow.f64 N/A

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

   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. lift-neg.f64 95.3

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

6. Applied rewrites 95.3%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 95.3%

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

10. Final simplification 95.3%

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

Alternative 7: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-exp.f64 N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 8: 94.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* (* x x) 10.0)) (sin (fma (PI) 0.5 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-cos.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. exp-prod N/A

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

   10. pow2 N/A

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

   11. pow-unpow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lift-cos.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-+.f64 98.2

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 98.2%

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

    1. lift-pow.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. pow-pow N/A

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

    4. pow2 N/A

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

    5. lift-exp.f64 N/A

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

    6. pow-exp N/A

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

    7. lower-exp.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 94.5

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

11. Applied rewrites 94.5%

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

Alternative 9: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 10: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

5. Applied rewrites 27.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 27.6

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

7. Applied rewrites 27.6%

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

Alternative 11: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 27.6%

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

7. Applied rewrites 27.6%

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

Alternative 12: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. Step-by-step derivation

7. Applied rewrites 21.3%

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

Alternative 13: 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

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 14: 9.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 9.8%

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

Alternative 15: 9.7% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-exp.f64 N/A

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

   5. exp-prod N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.7%

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

14. Final simplification 9.7%

    \[\leadsto \left(\left(x \cdot x\right) \cdot -0.5\right) \cdot 1 \]
15. Add Preprocessing

Alternative 16: 1.5% accurate, 216.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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. Final simplification 1.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(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)))))