exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 6

Speedup: 1.0×

• 49.5% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

3. Simplified 99.9%

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

   1. difference-of-sqr--1 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. exp-prod 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.94:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.94) (exp -1.0) (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.94) {
		tmp = exp(-1.0);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.94d0) then
        tmp = exp((-1.0d0))
    else
        tmp = exp((x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.94) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = Math.exp((x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.94:
		tmp = math.exp(-1.0)
	else:
		tmp = math.exp((x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.94)
		tmp = exp(-1.0);
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.94)
		tmp = exp(-1.0);
	else
		tmp = exp((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.94], N[Exp[-1.0], $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.93999999999999995

   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 98.8%

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

   if 0.93999999999999995 < (*.f64 x x)

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. unpow2 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.94:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

3. Simplified 99.9%

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

   1. difference-of-sqr--1 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 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 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 51.3% 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 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.6%

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

5. Final simplification 50.6%

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

Alternative 6: 10.5% accurate, 106.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{x \cdot x + -1} \cdot \sqrt[3]{x \cdot x + -1}\right) \cdot \sqrt[3]{x \cdot x + -1}}} \]

   2. exp-prod 100.0%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{x \cdot x + -1} \cdot \sqrt[3]{x \cdot x + -1}}\right)}^{\left(\sqrt[3]{x \cdot x + -1}\right)}} \]

   3. pow2 100.0%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{x \cdot x + -1}\right)}^{2}}}\right)}^{\left(\sqrt[3]{x \cdot x + -1}\right)} \]

   4. fma-def 100.0%

      \[\leadsto {\left(e^{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{x \cdot x + -1}\right)} \]

   5. fma-def 100.0%

      \[\leadsto {\left(e^{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}} \]

6. Taylor expanded in x around inf 10.4%

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

7. Final simplification 10.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))