expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 62.4%

Time: 16.1s

Alternatives: 2

Speedup: 505.0×

• could not determine a ground truth

  1. x = -2.346369712811804e+146

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: 6.6% 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: 62.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-309}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5e-309) 1.0 (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-309) {
		tmp = 1.0;
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-309) then
        tmp = 1.0d0
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-309) {
		tmp = 1.0;
	} else {
		tmp = Math.exp(-x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-309:
		tmp = 1.0
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-309)
		tmp = 1.0;
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-309)
		tmp = 1.0;
	else
		tmp = exp(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-309], 1.0, N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999995e-309

   1. Initial program 8.4%

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

      1. exp-neg 8.4%

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

      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. Step-by-step derivation

      1. add-exp-log 8.5%

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

      2. div-exp 8.5%

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

   5. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 8.5%

         \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right) \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}} \]

      2. pow3 8.5%

         \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]

   7. Applied egg-rr 8.5%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]

   8. Taylor expanded in x around inf 99.0%

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

   if 4.9999999999999995e-309 < x

   1. Initial program 4.7%

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

      1. exp-neg 4.7%

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

      2. associate-*r/ 4.7%

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

      3. *-rgt-identity 4.7%

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

   3. Simplified 4.7%

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

      1. add-exp-log 4.7%

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

      2. div-exp 4.7%

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

   5. Applied egg-rr 4.7%

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

   6. Taylor expanded in x around inf 38.4%

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

      1. neg-mul-1 38.4%

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

   8. Simplified 38.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-309}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2: 42.2% accurate, 505.0× 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 6.1%

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

   1. exp-neg 6.1%

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

   2. associate-*r/ 6.1%

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

   3. *-rgt-identity 6.1%

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

3. Simplified 6.1%

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

   1. add-exp-log 6.1%

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

   2. div-exp 6.1%

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

5. Applied egg-rr 6.1%

   \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 6.1%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right) \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}} \]

   2. pow3 6.1%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]

7. Applied egg-rr 6.1%

   \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]

8. Taylor expanded in x around inf 39.1%

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

9. Final simplification 39.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023207 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))