exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 4

Speedup: 1.0×

• 50.5% 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

real(8) function code(x)
    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

real(8) function code(x)
    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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 75.4%

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

   1. lift-/.f64 N/A

      \[\leadsto e^{\color{blue}{\frac{-1 + {x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}}} \]

   2. lift-+.f64 N/A

      \[\leadsto e^{\frac{\color{blue}{-1 + {x}^{4}}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\frac{\color{blue}{{x}^{4} + -1}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   4. metadata-eval N/A

      \[\leadsto e^{\frac{{x}^{4} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto e^{\frac{\color{blue}{{x}^{4} - 1}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   6. lift-pow.f64 N/A

      \[\leadsto e^{\frac{\color{blue}{{x}^{4}} - 1}{\mathsf{fma}\left(x, x, 1\right)}} \]

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. metadata-eval N/A

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

   12. lift-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. flip-+ N/A

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

   16. difference-of-sqr--1 N/A

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

   17. *-commutative N/A

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

   18. distribute-rgt-in N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 0.95:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (* x_m x_m) 0.95) (exp -1.0) (exp (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 0.95) {
		tmp = exp(-1.0);
	} else {
		tmp = exp((x_m * x_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((x_m * x_m) <= 0.95d0) then
        tmp = exp((-1.0d0))
    else
        tmp = exp((x_m * x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if ((x_m * x_m) <= 0.95) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = Math.exp((x_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if (x_m * x_m) <= 0.95:
		tmp = math.exp(-1.0)
	else:
		tmp = math.exp((x_m * x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 0.95)
		tmp = exp(-1.0);
	else
		tmp = exp(Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 0.95)
		tmp = exp(-1.0);
	else
		tmp = exp((x_m * x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 0.95], N[Exp[-1.0], $MachinePrecision], N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.4%

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

   if 0.94999999999999996 < (*.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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. 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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto e^{0 - \color{blue}{\left(1 - x \cdot x\right)}} \]

   4. associate--r- N/A

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 4: 50.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ e^{-1} \end{array} \]

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = exp((-1.0d0))
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.exp(-1.0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[-1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.6%

   \[\leadsto e^{\color{blue}{-1}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024327 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))