exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 7

Speedup: 1.0×

• 49.0% 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, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ e^{\mathsf{fma}\left(\sqrt{x\_m} \cdot \sqrt{x\_m}, x\_m, -1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (exp (fma (* (sqrt x_m) (sqrt x_m)) x_m -1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return exp(fma((sqrt(x_m) * sqrt(x_m)), x_m, -1.0));
}

Julia

x_m = abs(x)
function code(x_m)
	return exp(fma(Float64(sqrt(x_m) * sqrt(x_m)), x_m, -1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(N[(N[Sqrt[x$95$m], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision] * x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 100.0%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 47.3

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

6. Applied rewrites 47.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, 1\right) \cdot \frac{\mathsf{fma}\left(x\_m, x\_m, -1\right)}{\mathsf{fma}\left(x\_m, x\_m, -1\right) \cdot \mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5)
   (* (fma x_m x_m 1.0) (/ (fma x_m x_m -1.0) (* (fma x_m x_m -1.0) (E))))
   (exp (* x_m x_m))))

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 88.9%

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

   9. Applied rewrites 73.5%

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

   if 1.5 < x

   1. Initial program 100.0%

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

      1. rem-square-sqrt N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. mul-1-neg N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-neg-rev N/A

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

      10. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (exp (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(x$95$m * x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 100.0%

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

Alternative 4: 88.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-x\_m, x\_m, 1\right)\\ \mathbf{if}\;1 - x\_m \cdot x\_m \leq -\infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(x\_m, x\_m, 1\right)}{t\_0 \cdot \mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (- x_m) x_m 1.0)))
   (if (<= (- 1.0 (* x_m x_m)) (- INFINITY))
     (/ (* x_m x_m) (E))
     (/ (* t_0 (fma x_m x_m 1.0)) (* t_0 (E))))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -inf.0

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.4%

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

   9. Applied rewrites 76.4%

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

   11. Applied rewrites 89.4%

       \[\leadsto \frac{\mathsf{fma}\left(-x, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\mathsf{fma}\left(-x, x, 1\right) \cdot \mathsf{E}\left(\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0) (/ 1.0 (E)) (/ (* x_m x_m) (E))))

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.3%

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

   if 1 < x

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 57.0%

       \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{E}\left(\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\mathsf{fma}\left(x\_m, x\_m, 1\right)}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (fma x_m x_m 1.0) (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-exp.f64 82.0

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

5. Applied rewrites 82.0%

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

7. Applied rewrites 82.0%

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

Alternative 7: 51.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-exp.f64 82.0

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

5. Applied rewrites 82.0%

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 58.2%

    \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))