exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 11

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{e}{{\left(e^{2 \cdot x}\right)}^{\left(0.5 \cdot x\right)}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ E (pow (exp (* 2.0 x)) (* 0.5 x)))))

C

double code(double x) {
	return 1.0 / (((double) M_E) / pow(exp((2.0 * x)), (0.5 * x)));
}

Java

public static double code(double x) {
	return 1.0 / (Math.E / Math.pow(Math.exp((2.0 * x)), (0.5 * x)));
}

Python

def code(x):
	return 1.0 / (math.e / math.pow(math.exp((2.0 * x)), (0.5 * x)))

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / (exp(Float64(2.0 * x)) ^ Float64(0.5 * x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (2.71828182845904523536 / (exp((2.0 * x)) ^ (0.5 * x)));
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[Power[N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision], N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

    \[\leadsto \frac{1}{\frac{e}{{\left(e^{2 \cdot x}\right)}^{\color{blue}{\left(0.5 \cdot x\right)}}}} \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{e}{{\left(e^{x}\right)}^{x}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(E / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{e}{{\left(e^{x}\right)}^{x}}}} \]
4. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-E.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-exp N/A

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

   7. pow2 N/A

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

   8. e-exp-1 N/A

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

   9. div-exp N/A

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

   10. pow2 N/A

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

   11. exp-neg N/A

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

   12. sqr-neg-rev N/A

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

   13. fp-cancel-sign-sub N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. pow2 N/A

       \[\leadsto e^{\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right)\right)\right)} \]

   16. distribute-neg-in N/A

       \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right)}} \]

   17. metadata-eval N/A

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

   18. mul-1-neg N/A

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

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

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

   20. metadata-eval N/A

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

   21. *-lft-identity N/A

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

5. Applied rewrites 100.0%

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

Alternative 4: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1)
   (/
    1.0
    (/
     E
     (fma
      (fma (fma 0.16666666666666666 (* x x) 0.5) (* x x) 1.0)
      (* x x)
      1.0)))
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.1) {
		tmp = 1.0 / (((double) M_E) / fma(fma(fma(0.16666666666666666, (x * x), 0.5), (x * x), 1.0), (x * x), 1.0));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1)
		tmp = Float64(1.0 / Float64(exp(1) / fma(fma(fma(0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), 1.0), Float64(x * x), 1.0)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.1], N[(1.0 / N[(E / N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   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. lift-*.f64 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. exp-diff N/A

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

      9. lower-/.f64 N/A

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

      10. exp-1-e N/A

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

      11. lower-E.f64 N/A

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

      12. pow2 N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-exp.f64 100.0

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

   3. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. unpow-prod-down N/A

         \[\leadsto \frac{1}{\frac{e}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \frac{1}{\frac{e}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\frac{e}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + \color{blue}{1}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 1\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 1\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      13. pow2 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

      16. lift-*.f64 94.5

          \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

   8. Applied rewrites 94.5%

      \[\leadsto \frac{1}{\frac{e}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}} \]

   if 2.10000000000000009 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return exp(fma(x, x, -1.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto e^{{x}^{2} - 1 \cdot \color{blue}{1}} \]

   2. fp-cancel-sub-sign-inv N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 6: 91.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   E
   (fma (fma (fma 0.16666666666666666 (* x x) 0.5) (* x x) 1.0) (* x x) 1.0))))

C

double code(double x) {
	return 1.0 / (((double) M_E) / fma(fma(fma(0.16666666666666666, (x * x), 0.5), (x * x), 1.0), (x * x), 1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / fma(fma(fma(0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), 1.0), Float64(x * x), 1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. unpow-prod-down N/A

      \[\leadsto \frac{1}{\frac{e}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   2. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{e}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\frac{e}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + \color{blue}{1}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\frac{e}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 1\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 1\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   13. pow2 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   15. pow2 N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

   16. lift-*.f64 91.8

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

8. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\frac{e}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}} \]
9. Add Preprocessing

Alternative 7: 88.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ E (fma (* (fma (* 0.5 x) x 1.0) x) x 1.0))))

C

double code(double x) {
	return 1.0 / (((double) M_E) / fma((fma((0.5 * x), x, 1.0) * x), x, 1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / fma(Float64(fma(Float64(0.5 * x), x, 1.0) * x), x, 1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[(N[(N[(N[(0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. unpow-prod-down N/A

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

   2. sqr-pow N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 88.0

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

8. Applied rewrites 88.0%

   \[\leadsto \frac{1}{\frac{e}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, 1\right)}}} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{2} + 1, x \cdot x, 1\right)}} \]

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{2} + 1\right) \cdot x, x, 1\right)}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + 1\right) \cdot x, x, 1\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(x \cdot \left(\frac{1}{2} \cdot x\right) + 1\right) \cdot x, x, 1\right)}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(\left(x \cdot \frac{1}{2}\right) \cdot x + 1\right) \cdot x, x, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot x + 1\right) \cdot x, x, 1\right)}} \]

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 88.0

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, x, 1\right) \cdot x, x, 1\right)}} \]

10. Applied rewrites 88.0%

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

Alternative 8: 87.6% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ E (fma (* (* x x) 0.5) (* x x) 1.0))))

C

double code(double x) {
	return 1.0 / (((double) M_E) / fma(((x * x) * 0.5), (x * x), 1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / fma(Float64(Float64(x * x) * 0.5), Float64(x * x), 1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. unpow-prod-down N/A

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

   2. sqr-pow N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 88.0

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

8. Applied rewrites 88.0%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 87.6

       \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, x \cdot x, 1\right)}} \]

11. Applied rewrites 87.6%

    \[\leadsto \frac{1}{\frac{e}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, \color{blue}{x} \cdot x, 1\right)}} \]
12. Add Preprocessing

Alternative 9: 75.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1:\\ \;\;\;\;\frac{x \cdot x}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1.0) {
		tmp = (x * x) / ((double) M_E);
	} else {
		tmp = 1.0 / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1.0) {
		tmp = (x * x) / Math.E;
	} else {
		tmp = 1.0 / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -1.0:
		tmp = (x * x) / math.e
	else:
		tmp = 1.0 / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -1.0)
		tmp = Float64(Float64(x * x) / exp(1));
	else
		tmp = Float64(1.0 / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -1.0)
		tmp = (x * x) / 2.71828182845904523536;
	else
		tmp = 1.0 / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in x around 0

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

      1. exp-neg N/A

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

      2. pow2 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      3. div-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      4. e-exp-1 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      5. pow2 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      6. pow-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. e-exp-1 N/A

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

      10. metadata-eval N/A

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

      11. rec-exp N/A

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

      12. e-exp-1 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

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

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

      19. *-rgt-identity N/A

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

      20. mul-1-neg N/A

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2}}{e} \]
   9. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 51.8

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

   10. Applied rewrites 51.8%

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

   if -1 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   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. lift-*.f64 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. exp-diff N/A

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

      9. lower-/.f64 N/A

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

      10. exp-1-e N/A

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

      11. lower-E.f64 N/A

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

      12. pow2 N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-exp.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]
   5. Step-by-step derivation

      1. lift-E.f64 98.9

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

   6. Applied rewrites 98.9%

      \[\leadsto \frac{1}{\color{blue}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right)}{e} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.1%

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

5. Taylor expanded in x around 0

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

   1. exp-neg N/A

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

   2. pow2 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   3. div-exp N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   4. e-exp-1 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   5. pow2 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   6. pow-exp N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. e-exp-1 N/A

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. e-exp-1 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. mul-1-neg N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

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

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

   19. *-rgt-identity N/A

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

   20. mul-1-neg N/A

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

7. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
8. Add Preprocessing

Alternative 11: 51.1% accurate, 9.3× speedup

Math

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

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. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]
5. Step-by-step derivation

   1. lift-E.f64 51.1

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

6. Applied rewrites 51.1%

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

Reproduce

herbie shell --seed 2025096 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))