exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 11

Speedup: 1.0×

• 50.1% 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

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

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

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. 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 2: 87.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 100.0%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.3%

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

   10. Applied rewrites 75.3%

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{x}{e} \cdot 0.5\right) \cdot x\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

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

Alternative 3: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-8) (/ (fma (* x x) E E) (* E E)) (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-8) {
		tmp = fma((x * x), ((double) M_E), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1e-8)
		tmp = Float64(fma(Float64(x * x), exp(1), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-8], N[(N[(N[(x * x), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-8

   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. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 100.0%

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

   if 1e-8 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 93.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot t\_0\right) \cdot \left(x \cdot x\right), x \cdot x, -1\right)}{\mathsf{fma}\left(t\_0, x \cdot x, -1\right) \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{x}{e} \cdot 0.5\right) \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.5 1.0)))
   (if (<= (* x x) 2e+149)
     (/
      (fma (* (* t_0 t_0) (* x x)) (* x x) -1.0)
      (* (fma t_0 (* x x) -1.0) E))
     (* (* (* (* (/ x E) 0.5) x) x) x))))

C

double code(double x) {
	double t_0 = fma((x * x), 0.5, 1.0);
	double tmp;
	if ((x * x) <= 2e+149) {
		tmp = fma(((t_0 * t_0) * (x * x)), (x * x), -1.0) / (fma(t_0, (x * x), -1.0) * ((double) M_E));
	} else {
		tmp = ((((x / ((double) M_E)) * 0.5) * x) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (Float64(x * x) <= 2e+149)
		tmp = Float64(fma(Float64(Float64(t_0 * t_0) * Float64(x * x)), Float64(x * x), -1.0) / Float64(fma(t_0, Float64(x * x), -1.0) * exp(1)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x / exp(1)) * 0.5) * x) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+149], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * E), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / E), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e149

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 91.6%

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

   if 2.0000000000000001e149 < (*.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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{x}{e} \cdot 0.5\right) \cdot x\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\right) \cdot \left(x \cdot x\right), x \cdot x, -1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.5, 1\right), x \cdot x, -1\right) \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{x}{e} \cdot 0.5\right) \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{x}{e} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-8)
   (/ (fma (* x x) E E) (* E E))
   (* (* (* (* (/ x E) (* x x)) x) (* x x)) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-8) {
		tmp = fma((x * x), ((double) M_E), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = ((((x / ((double) M_E)) * (x * x)) * x) * (x * x)) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1e-8)
		tmp = Float64(fma(Float64(x * x), exp(1), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x / exp(1)) * Float64(x * x)) * x) * Float64(x * x)) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-8], N[(N[(N[(x * x), $MachinePrecision] * E + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / E), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-8

   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. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 100.0%

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

   if 1e-8 < (*.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

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.2%

      \[\leadsto \left(\left(0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(\frac{x \cdot x}{e} \cdot x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.2%

       \[\leadsto 0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(\frac{x \cdot x}{e} \cdot x\right) \cdot x\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 86.2%

       \[\leadsto 0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\frac{x}{e} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, e, e\right)}{e \cdot e}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{x}{e} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 93.0%

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

6. Final simplification 93.0%

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

Alternative 7: 87.8% accurate, 3.3× speedup

Math

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

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

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 87.4%

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

7. Applied rewrites 87.4%

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

Alternative 8: 75.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-8}:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 1e-8) (/ 1.0 E) (* (/ x E) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-8) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x / ((double) M_E)) * x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 1e-8) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x / Math.E) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1e-8:
		tmp = 1.0 / math.e
	else:
		tmp = (x / math.e) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1e-8)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x / exp(1)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 1e-8)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x / 2.71828182845904523536) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-8], N[(1.0 / E), $MachinePrecision], N[(N[(x / E), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. rec-exp N/A

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

      3. lower-/.f64 N/A

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

      4. exp-1-e N/A

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

      5. lower-E.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e-8 < (*.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. metadata-eval N/A

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

      6. rec-exp N/A

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

      7. lower-/.f64 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.2%

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

   10. Applied rewrites 51.2%

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

Alternative 9: 75.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
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. metadata-eval N/A

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

   6. rec-exp N/A

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

   7. lower-/.f64 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f64 75.2

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

5. Applied rewrites 75.2%

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

7. Applied rewrites 74.4%

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

9. Applied rewrites 75.2%

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

Alternative 10: 75.9% 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. 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. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. rec-exp N/A

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

   4. e-exp-1 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-E.f64 75.2

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

7. Applied rewrites 75.2%

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

Alternative 11: 50.6% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. rec-exp N/A

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

   3. lower-/.f64 N/A

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

   4. exp-1-e N/A

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

   5. lower-E.f64 50.6

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

5. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))