ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 97.6%

Time: 35.6s

Alternatives: 9

Speedup: 1.0×

Specification

\[\frac{199}{100} \leq x \land x \leq \frac{201}{100}\]

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (exp (* 10 (* 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 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (exp (* 10 (* 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 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (pow (pow (exp (+ x x)) x) 5)))

C

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

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 = cos(x) * ((exp((x + x)) ** x) ** 5.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision], x], $MachinePrecision], 5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 95.2%

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

3. Applied rewrites 95.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. exp-fabs N/A

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

   4. lift-exp.f64 N/A

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

   5. rem-sqrt-square-rev N/A

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

   6. sqrt-pow2 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left(e^{x \cdot x} \cdot e^{x \cdot x}\right)}^{\color{blue}{\left(3 + 2\right)}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x} \cdot e^{x \cdot x}\right)}^{\left(3 + 2\right)}} \]

   10. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(\color{blue}{e^{x \cdot x}} \cdot e^{x \cdot x}\right)}^{\left(3 + 2\right)} \]

   11. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{x \cdot x} \cdot \color{blue}{e^{x \cdot x}}\right)}^{\left(3 + 2\right)} \]

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

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\left(x \cdot x\right) \cdot 2}\right)}}^{\left(3 + 2\right)} \]

   13. lower-exp.f64 N/A

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\left(x \cdot x\right) \cdot 2}\right)}}^{\left(3 + 2\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\left(x \cdot x\right) \cdot 2}}\right)}^{\left(3 + 2\right)} \]

   15. metadata-eval 95.3%

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

5. Applied rewrites 95.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-lft-sqr N/A

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

   4. lift-*.f64 N/A

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

   5. exp-prod N/A

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

   6. lift-*.f64 N/A

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

   7. exp-prod N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f64 N/A

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

   10. prod-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-+.f64 97.6%

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

7. Applied rewrites 97.6%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (pow (pow (exp x) x) 10)))

C

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

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 = cos(x) * ((exp(x) ** x) ** 10.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision], 10], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 95.2%

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

3. Applied rewrites 95.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-prod N/A

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

   7. exp-fabs N/A

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

   8. lift-exp.f64 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. sqrt-pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-exp.f64 N/A

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

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

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

   15. lower-exp.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. mult-flip-rev N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 97.0%

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 96.7%

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

Alternative 3: 95.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (pow (exp (* x x)) 10)))

C

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

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 = cos(x) * (exp((x * x)) ** 10.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision], 10], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 95.2%

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

3. Applied rewrites 95.2%

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

Alternative 4: 95.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (pow (exp 10) (* 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], $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. 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. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 95.2%

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

3. Applied rewrites 95.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-neg-rev N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-prod N/A

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

   7. exp-fabs N/A

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

   8. lift-exp.f64 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. sqrt-pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-exp.f64 N/A

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

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

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

   15. lower-exp.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. mult-flip-rev N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 97.0%

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

5. Applied rewrites 97.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. pow-exp N/A

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

   5. pow-exp N/A

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

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

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

7. Applied rewrites 95.2%

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

Alternative 5: 94.5% accurate, 1.0× speedup

Math

\[\sin \left(\frac{1}{2} \cdot \pi + x\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  (* (sin (+ (* 1/2 PI) x)) (exp (* 10 (* x x)))))

C

double code(double x) {
	return sin(((0.5 * ((double) M_PI)) + x)) * exp((10.0 * (x * x)));
}

Java

public static double code(double x) {
	return Math.sin(((0.5 * Math.PI) + x)) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.sin(((0.5 * math.pi) + x)) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(sin(Float64(Float64(0.5 * pi) + x)) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = sin(((0.5 * pi) + x)) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Sin[N[(N[(1/2 * Pi), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(10 * 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. 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. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. mult-flip N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-PI.f64 94.5%

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

3. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \pi + x\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. Add Preprocessing

Alternative 6: 11.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (pow (+ 1 x) (* 10 x))))

C

double code(double x) {
	return cos(x) * pow((1.0 + x), (10.0 * 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) * ((1.0d0 + x) ** (10.0d0 * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[(1 + x), $MachinePrecision], N[(10 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 95.2%

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

3. Applied rewrites 95.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative 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. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-*.f64 94.7%

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 11.4%

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

8. Applied rewrites 11.4%

   \[\leadsto \cos x \cdot {\color{blue}{\left(1 + x\right)}}^{\left(10 \cdot x\right)} \]
9. Add Preprocessing

Alternative 7: 9.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (cos x) (+ 1 (* 10 (pow x 2)))))

C

double code(double x) {
	return cos(x) * (1.0 + (10.0 * pow(x, 2.0)));
}

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 = cos(x) * (1.0d0 + (10.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * (1.0 + (10.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.cos(x) * (1.0 + (10.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return Float64(cos(x) * Float64(1.0 + Float64(10.0 * (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (1.0 + (10.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[(1 + N[(10 * N[Power[x, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 9.8%

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

4. Applied rewrites 9.8%

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

Alternative 8: 1.5% accurate, 15.4× speedup

Math

\[1 + \left(\frac{19}{2} \cdot x\right) \cdot x \]

FPCore

(FPCore (x)
  :precision binary64
  (+ 1 (* (* 19/2 x) x)))

C

double code(double x) {
	return 1.0 + ((9.5 * 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 = 1.0d0 + ((9.5d0 * x) * x)
end function

Java

public static double code(double x) {
	return 1.0 + ((9.5 * x) * x);
}

Python

def code(x):
	return 1.0 + ((9.5 * x) * x)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(9.5 * x) * x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((9.5 * x) * x);
end

Wolfram

code[x_] := N[(1 + N[(N[(19/2 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{19}{2} \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + \color{blue}{\frac{19}{2} \cdot {x}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 + \frac{19}{2} \cdot \color{blue}{{x}^{2}} \]

   3. lower-pow.f64 1.5%

      \[\leadsto 1 + \frac{19}{2} \cdot {x}^{\color{blue}{2}} \]

4. Applied rewrites 1.5%

   \[\leadsto \color{blue}{1 + \frac{19}{2} \cdot {x}^{2}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 1 + \frac{19}{2} \cdot \color{blue}{{x}^{2}} \]

   2. lift-pow.f64 N/A

      \[\leadsto 1 + \frac{19}{2} \cdot {x}^{\color{blue}{2}} \]

   3. pow2 N/A

      \[\leadsto 1 + \frac{19}{2} \cdot \left(x \cdot \color{blue}{x}\right) \]

   4. associate-*r* N/A

      \[\leadsto 1 + \left(\frac{19}{2} \cdot x\right) \cdot \color{blue}{x} \]

   5. metadata-eval N/A

      \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{-19}{2}\right)\right) \cdot x\right) \cdot x \]

   6. metadata-eval N/A

      \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right)\right)\right) \cdot x\right) \cdot x \]

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

      \[\leadsto 1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot x\right)\right) \cdot x \]

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

      \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

   9. lift-neg.f64 N/A

      \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot \left(-x\right)\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot \left(-x\right)\right) \cdot \color{blue}{x} \]

   11. lift-neg.f64 N/A

       \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right) \cdot x \]

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

       \[\leadsto 1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right) \cdot x\right)\right) \cdot x \]

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

       \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{19}{2}\right)\right)\right)\right) \cdot x\right) \cdot x \]

   14. metadata-eval N/A

       \[\leadsto 1 + \left(\left(\mathsf{neg}\left(\frac{-19}{2}\right)\right) \cdot x\right) \cdot x \]

   15. metadata-eval N/A

       \[\leadsto 1 + \left(\frac{19}{2} \cdot x\right) \cdot x \]

   16. lower-*.f64 1.5%

       \[\leadsto 1 + \left(\frac{19}{2} \cdot x\right) \cdot x \]

6. Applied rewrites 1.5%

   \[\leadsto 1 + \left(\frac{19}{2} \cdot x\right) \cdot \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 1"
  :precision binary64
  :pre (and (<= 199/100 x) (<= x 201/100))
  (* (cos x) (exp (* 10 (* x x)))))