exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 4

Speedup: 1.0×

• 50.0% 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

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

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. Step-by-step derivation

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

   7. fma-define 100.0%

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

3. Simplified 100.0%

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

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. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto e^{-1 + x \cdot x} \]
6. Add Preprocessing

Alternative 3: 75.3% accurate, 8.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 1.0) (/ 1.0 E) (* x (/ x E))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   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. sqr-neg 100.0%

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

      3. associate--r- 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.4%

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

      1. distribute-rgt1-in 98.4%

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

      2. unpow2 98.4%

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

      3. fma-undefine 98.4%

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

      4. metadata-eval 98.4%

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

      5. rec-exp 98.4%

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

      6. e-exp-1 98.4%

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

      7. associate-*r/ 98.4%

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

      8. metadata-eval 98.4%

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

      9. distribute-rgt-neg-in 98.4%

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

      10. *-commutative 98.4%

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

      11. neg-mul-1 98.4%

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

      12. remove-double-neg 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 97.8%

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

   if 1 < (*.f64 x x)

   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. sqr-neg 100.0%

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

      3. associate--r- 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 52.7%

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

      1. distribute-rgt1-in 52.7%

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

      2. unpow2 52.7%

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

      3. fma-undefine 52.7%

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

      4. metadata-eval 52.7%

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

      5. rec-exp 52.7%

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

      6. e-exp-1 52.7%

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

      7. associate-*r/ 52.7%

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

      8. metadata-eval 52.7%

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

      9. distribute-rgt-neg-in 52.7%

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

      10. *-commutative 52.7%

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

      11. neg-mul-1 52.7%

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

      12. remove-double-neg 52.7%

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

   7. Simplified 52.7%

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

   8. Taylor expanded in x around inf 52.7%

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

      1. unpow2 52.7%

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

      2. associate-/l* 52.7%

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

   10. Applied egg-rr 52.7%

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

Alternative 4: 50.8% accurate, 35.3× 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. Step-by-step derivation

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 74.1%

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

   1. distribute-rgt1-in 74.1%

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

   2. unpow2 74.1%

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

   3. fma-undefine 74.1%

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

   4. metadata-eval 74.1%

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

   5. rec-exp 74.1%

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

   6. e-exp-1 74.1%

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

   7. associate-*r/ 74.1%

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

   8. metadata-eval 74.1%

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

   9. distribute-rgt-neg-in 74.1%

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

   10. *-commutative 74.1%

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

   11. neg-mul-1 74.1%

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

   12. remove-double-neg 74.1%

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

7. Simplified 74.1%

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

8. Taylor expanded in x around 0 47.5%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))