exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 1.3s

Alternatives: 3

Speedup: 1.0×

• 48.9% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (expm1 (fma x x -1.0))))

C

double code(double x) {
	return 1.0 + expm1(fma(x, x, -1.0));
}

Julia

function code(x)
	return Float64(1.0 + expm1(fma(x, x, -1.0)))
end

Wolfram

code[x_] := N[(1.0 + N[(Exp[N[(x * x + -1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. associate--r- 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot x + -1\right)\right)}} \]

   2. log1p-udef 100.0%

      \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot x + -1\right)\right)}} \]

   3. add-exp-log 100.0%

      \[\leadsto \color{blue}{1 + \mathsf{expm1}\left(x \cdot x + -1\right)} \]

   4. fma-def 100.0%

      \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]

6. Final simplification 100.0%

   \[\leadsto 1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-1 + x \cdot x} \end{array} \]

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(-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]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. associate--r- 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[-1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. associate--r- 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 55.5%

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

5. Final simplification 55.5%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))