exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 1.8s

Alternatives: 6

Speedup: 0.9×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[e^{-\left(1 - x \cdot x\right)} \]

FPCore

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

C

double code(double x) {
	return exp(-(1.0 - (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 = exp(-(1.0d0 - (x * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[e^{-\left(1 - x \cdot x\right)} \]

FPCore

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

C

double code(double x) {
	return exp(-(1.0 - (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 = exp(-(1.0d0 - (x * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\frac{e^{x \cdot x}}{e} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (exp (* x x)) E))

C

double code(double x) {
	return exp((x * x)) / ((double) M_E);
}

Java

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

Python

def code(x):
	return math.exp((x * x)) / math.e

Julia

function code(x)
	return Float64(exp(Float64(x * x)) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) / 2.71828182845904523536;
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. exp-1-e N/A

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   9. lower-E.f64 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[e^{x \cdot x - 1} \]

FPCore

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

C

double code(double x) {
	return exp(((x * x) - 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 = exp(((x * x) - 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   4. lower--.f64 100.0%

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

3. Applied rewrites 100.0%

   \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]
4. Add Preprocessing

Alternative 3: 93.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \frac{1 + \sqrt{\sqrt{t\_0 \cdot t\_0}}}{e} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (* x x) x) x)))
  (/ (+ 1.0 (sqrt (sqrt (* t_0 t_0)))) E)))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return (1.0 + sqrt(sqrt((t_0 * t_0)))) / ((double) M_E);
}

Java

public static double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return (1.0 + Math.sqrt(Math.sqrt((t_0 * t_0)))) / Math.E;
}

Python

def code(x):
	t_0 = ((x * x) * x) * x
	return (1.0 + math.sqrt(math.sqrt((t_0 * t_0)))) / math.e

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	return Float64(Float64(1.0 + sqrt(sqrt(Float64(t_0 * t_0)))) / exp(1))
end

MATLAB

function tmp = code(x)
	t_0 = ((x * x) * x) * x;
	tmp = (1.0 + sqrt(sqrt((t_0 * t_0)))) / 2.71828182845904523536;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1.0 + N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. exp-1-e N/A

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   9. lower-E.f64 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 76.0%

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

6. Applied rewrites 76.0%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \frac{1 + x \cdot \color{blue}{x}}{e} \]

   3. fabs-sqr N/A

      \[\leadsto \frac{1 + \left|x \cdot x\right|}{e} \]

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

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   6. lower-unsound-*.f32 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   8. lower-unsound-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   10. lower-*.f64 87.3%

       \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

8. Applied rewrites 87.3%

   \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]
9. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}}{e} \]

   2. sqrt-unprod N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   4. lower-*.f64 93.6%

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   7. associate-*l* N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   8. *-commutative N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   10. cube-unmult N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   11. lower-pow.f32 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   12. lower-unsound-pow.f32 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   14. lower-unsound-pow.f32 97.7%

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left( {x}^{3} \right)_{\text{binary32}} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   15. lower-pow.f32 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   16. pow3 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   18. lower-*.f64 93.6%

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}{e} \]

   21. associate-*l* N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}}{e} \]

   22. *-commutative N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)}}}{e} \]

   23. lift-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)}}}{e} \]

   24. cube-unmult N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}}}{e} \]

   25. lower-pow.f32 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}}}{e} \]

   26. lower-unsound-pow.f32 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}}}{e} \]

   27. lower-*.f64 N/A

       \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}}}{e} \]

10. Applied rewrites 93.6%

    \[\leadsto \frac{1 + \sqrt{\sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)}}}{e} \]
11. Add Preprocessing

Alternative 4: 87.3% accurate, 2.8× speedup

Math

\[\frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (+ 1.0 (sqrt (* (* x x) (* x x)))) E))

C

double code(double x) {
	return (1.0 + sqrt(((x * x) * (x * x)))) / ((double) M_E);
}

Java

public static double code(double x) {
	return (1.0 + Math.sqrt(((x * x) * (x * x)))) / Math.E;
}

Python

def code(x):
	return (1.0 + math.sqrt(((x * x) * (x * x)))) / math.e

Julia

function code(x)
	return Float64(Float64(1.0 + sqrt(Float64(Float64(x * x) * Float64(x * x)))) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + sqrt(((x * x) * (x * x)))) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[(1.0 + N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. exp-1-e N/A

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   9. lower-E.f64 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 76.0%

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

6. Applied rewrites 76.0%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \frac{1 + x \cdot \color{blue}{x}}{e} \]

   3. fabs-sqr N/A

      \[\leadsto \frac{1 + \left|x \cdot x\right|}{e} \]

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

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   6. lower-unsound-*.f32 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   8. lower-unsound-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

   10. lower-*.f64 87.3%

       \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]

8. Applied rewrites 87.3%

   \[\leadsto \frac{1 + \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{e} \]
9. Add Preprocessing

Alternative 5: 76.0% accurate, 5.6× speedup

Math

\[\frac{x \cdot x - -1}{e} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (- (* x x) -1.0) E))

C

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

Java

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

Python

def code(x):
	return ((x * x) - -1.0) / math.e

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - -1.0) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - -1.0) / 2.71828182845904523536;
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. exp-1-e N/A

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   9. lower-E.f64 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 76.0%

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

6. Applied rewrites 76.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

      \[\leadsto \frac{1 + x \cdot \color{blue}{x}}{e} \]

   4. +-commutative N/A

      \[\leadsto \frac{x \cdot x + \color{blue}{1}}{e} \]

   5. add-flip N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{x \cdot x - -1}{e} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{x \cdot x - \color{blue}{-1}}{e} \]

   8. lower-*.f64 76.0%

      \[\leadsto \frac{x \cdot x - -1}{e} \]

8. Applied rewrites 76.0%

   \[\leadsto \frac{x \cdot x - \color{blue}{-1}}{e} \]
9. Add Preprocessing

Alternative 6: 50.8% accurate, 9.3× speedup

Math

\[\frac{1}{e} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 1.0 E))

C

double code(double x) {
	return 1.0 / ((double) M_E);
}

Java

public static double code(double x) {
	return 1.0 / Math.E;
}

Python

def code(x):
	return 1.0 / math.e

Julia

function code(x)
	return Float64(1.0 / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate N/A

      \[\leadsto e^{\color{blue}{x \cdot x - 1}} \]

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. exp-1-e N/A

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   9. lower-E.f64 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e}} \]

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{e} \]
5. Step-by-step derivation

6. Applied rewrites 50.8%

   \[\leadsto \frac{\color{blue}{1}}{e} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))