exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 12

Speedup: 1.0×

• 49.1% 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

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

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

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

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

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

   14. pow2 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

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

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

   19. metadata-eval N/A

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

   20. *-lft-identity N/A

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

   21. +-commutative N/A

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

   22. pow2 N/A

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

5. Applied rewrites 100.0%

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

Alternative 3: 76.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(-(1.0 - (x * x))) <= 0.5) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(-(1.0 - (x * x))) <= 0.5:
		tmp = 1.0 / math.e
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(-(1.0 - (x * x))) <= 0.5)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   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 99.1

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

   6. Applied rewrites 99.1%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 52.8

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

   4. Applied rewrites 52.8%

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

      1. lift-exp.f64 N/A

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

      2. metadata-eval N/A

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

      3. rec-exp N/A

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

      4. e-exp-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-E.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2}}{\color{blue}{\mathsf{E}\left(\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

      4. lift-E.f64 52.8

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

   9. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-E.f64 N/A

          \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{x}{\mathsf{E}\left(\right)} \]

       7. lift-E.f64 52.8

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

   11. Applied rewrites 52.8%

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

Alternative 4: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) * exp(Float64(Float64(-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[Exp[N[((-x) * 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-E.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-exp N/A

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

   6. pow2 N/A

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

   7. e-exp-1 N/A

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

   8. div-exp N/A

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

   9. pow2 N/A

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

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

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

   11. exp-sum N/A

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

   12. e-exp-1 N/A

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

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

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

   14. pow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-E.f64 N/A

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

   17. lower-exp.f64 N/A

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

   18. pow2 N/A

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

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

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

   20. lower-*.f64 N/A

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

   21. lower-neg.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 5: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(0.5, e, e \cdot -0.3333333333333333\right), 0.5 \cdot e\right), x \cdot x, -e\right), x \cdot x, e\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (/
    1.0
    (fma
     (fma
      (fma (* (- x) x) (fma 0.5 E (* E -0.3333333333333333)) (* 0.5 E))
      (* x x)
      (- E))
     (* x x)
     E))
   (exp (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = 1.0 / fma(fma(fma((-x * x), fma(0.5, ((double) M_E), (((double) M_E) * -0.3333333333333333)), (0.5 * ((double) M_E))), (x * x), -((double) M_E)), (x * x), ((double) M_E));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(1.0 / fma(fma(fma(Float64(Float64(-x) * x), fma(0.5, exp(1), Float64(exp(1) * -0.3333333333333333)), Float64(0.5 * exp(1))), Float64(x * x), Float64(-exp(1))), Float64(x * x), exp(1)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(1.0 / N[(N[(N[(N[((-x) * x), $MachinePrecision] * N[(0.5 * E + N[(E * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(0.5 * E), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + (-E)), $MachinePrecision] * N[(x * x), $MachinePrecision] + E), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   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

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

   6. Applied rewrites 67.7%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(0.5, e, e \cdot -0.3333333333333333\right), 0.5 \cdot e\right), x \cdot x, -e\right), x \cdot x, e\right)}} \]

   if 1.21999999999999997 < 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 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 6: 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 7: 92.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(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] / 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. fp-cancel-sub-sign-inv N/A

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

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

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

   14. pow2 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

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

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

   19. metadata-eval N/A

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

   20. *-lft-identity N/A

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

   21. +-commutative N/A

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

   22. pow2 N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 92.3

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

8. Applied rewrites 92.3%

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

Alternative 8: 88.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * x), $MachinePrecision] * x + 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. 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. fp-cancel-sub-sign-inv N/A

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

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

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

   14. pow2 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

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

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

   19. metadata-eval N/A

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

   20. *-lft-identity N/A

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

   21. +-commutative N/A

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

   22. pow2 N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 88.2

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

8. Applied rewrites 88.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. pow2 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. *-rgt-identity N/A

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

   13. distribute-lft-in N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. pow2 N/A

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

   17. +-commutative N/A

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

   18. pow2 N/A

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

   19. associate-*r* N/A

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

10. Applied rewrites 88.2%

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

Alternative 9: 87.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * 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. 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. fp-cancel-sub-sign-inv N/A

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

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

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

   14. pow2 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

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

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

   19. metadata-eval N/A

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

   20. *-lft-identity N/A

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

   21. +-commutative N/A

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

   22. pow2 N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 87.9

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

11. Applied rewrites 87.9%

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

Alternative 10: 76.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-exp.f64 76.6

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

4. Applied rewrites 76.6%

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

   1. lift-exp.f64 N/A

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

   2. metadata-eval N/A

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

   3. rec-exp N/A

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

   4. e-exp-1 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-E.f64 76.6

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

6. Applied rewrites 76.6%

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

Alternative 11: 76.6% 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. 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

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

   1. div-add-rev N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-E.f64 76.6

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

6. Applied rewrites 76.6%

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

Alternative 12: 51.8% 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 51.8

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

6. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))