expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 61.4%

Time: 23.2s

Alternatives: 3

Speedup: 5.0×

• could not determine a ground truth

  1. x = -5.807403826626458e+261

Specification

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 61.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-16) (exp (- (log (fmod (exp x) 1.0)) x)) (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-16) {
		tmp = exp((log(fmod(exp(x), 1.0)) - x));
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-16)) then
        tmp = exp((log(mod(exp(x), 1.0d0)) - x))
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-16:
		tmp = math.exp((math.log(math.fmod(math.exp(x), 1.0)) - x))
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-16)
		tmp = exp(Float64(log(rem(exp(x), 1.0)) - x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-16], N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000004e-16

   1. Initial program 91.3%

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

      1. exp-neg 91.3%

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      2. associate-*r/ 91.7%

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      3. *-rgt-identity 91.7%

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Simplified 91.7%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   4. Taylor expanded in x around 0 91.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]
   5. Step-by-step derivation

      1. add-exp-log 91.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod 1\right)}}}{e^{x}} \]

      2. div-exp 91.9%

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

   6. Applied egg-rr 91.9%

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

   if -5.0000000000000004e-16 < x

   1. Initial program 4.4%

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

      1. exp-neg 4.4%

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      2. associate-*r/ 4.4%

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      3. *-rgt-identity 4.4%

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Simplified 4.4%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   4. Taylor expanded in x around 0 4.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]
   5. Step-by-step derivation

      1. add-exp-log 4.1%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod 1\right)}}}{e^{x}} \]

      2. div-exp 4.1%

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in x around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   9. Simplified 58.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2: 61.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-16) (/ (fmod (exp x) 1.0) (exp x)) (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-16) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-16)) then
        tmp = mod(exp(x), 1.0d0) / exp(x)
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-16:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-16)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000004e-16

   1. Initial program 91.3%

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

      1. exp-neg 91.3%

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      2. associate-*r/ 91.7%

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      3. *-rgt-identity 91.7%

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Simplified 91.7%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   4. Taylor expanded in x around 0 91.7%

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

   if -5.0000000000000004e-16 < x

   1. Initial program 4.4%

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

      1. exp-neg 4.4%

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      2. associate-*r/ 4.4%

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      3. *-rgt-identity 4.4%

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Simplified 4.4%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   4. Taylor expanded in x around 0 4.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]
   5. Step-by-step derivation

      1. add-exp-log 4.1%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod 1\right)}}}{e^{x}} \]

      2. div-exp 4.1%

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in x around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   9. Simplified 58.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 3: 60.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. exp-neg 8.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

   2. associate-*r/ 8.5%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

   3. *-rgt-identity 8.5%

      \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

3. Simplified 8.5%

   \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

4. Taylor expanded in x around 0 8.3%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]
5. Step-by-step derivation

   1. add-exp-log 8.3%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod 1\right)}}}{e^{x}} \]

   2. div-exp 8.3%

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

6. Applied egg-rr 8.3%

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

7. Taylor expanded in x around inf 57.8%

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

   1. neg-mul-1 57.8%

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

9. Simplified 57.8%

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

10. Final simplification 57.8%

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

Reproduce

herbie shell --seed 2023285 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))