exp neg sub

Average Error: 0.0 → 0.0

Time: 3.7s

Precision: binary64

Cost: 19456

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (* (pow (exp x) x) (exp -1.0)))

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

def code(x):
	return math.pow(math.exp(x), x) * math.exp(-1.0)

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

↓

function code(x)
	return Float64((exp(x) ^ x) * exp(-1.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (exp(x) ^ x) * exp(-1.0);
end

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   Proof
   (exp.f64 (+.f64 (*.f64 x x) -1)): 0 points increase in error, 0 points decrease in error
   (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 -1 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
   (exp.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (exp.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (exp.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	12992

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

Alternative 2
Error	0.0
Cost	6720

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

Alternative 3
Error	0.9
Cost	6464

\[e^{-1} \]

Alternative 4
Error	52.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))