expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 26.2%

Time: 10.4s

Alternatives: 4

Speedup: 1.9×

• could not determine a ground truth

  1. x = -4.4939025352825645e+259

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: 6.9% 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: 26.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(x \cdot x\right) \cdot -0.25\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (- 1.0 x) (fmod (exp x) (fma (* x x) -0.25 1.0)))
     (* (fmod 1.0 (* (* x x) -0.25)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = (1.0 - x) * fmod(exp(x), fma((x * x), -0.25, 1.0));
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(Float64(1.0 - x) * rem(exp(x), fma(Float64(x * x), -0.25, 1.0)));
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 8.8%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\frac{-1}{4} \cdot {x}^{2}}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 7.0%

      \[\leadsto \left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{-0.25}, 1\right)\right)\right) \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. unpow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. lower-*.f64 100.0

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{-0.25}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 26.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (- x -1.0) (fma (* x x) -0.25 1.0)) (exp (- x))))

C

double code(double x) {
	return fmod((x - -1.0), fma((x * x), -0.25, 1.0)) * exp(-x);
}

Julia

function code(x)
	return Float64(rem(Float64(x - -1.0), fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
4. Step-by-step derivation

5. Applied rewrites 27.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + 1\right)\right) \cdot e^{-x} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, 1\right)\right)}\right) \cdot e^{-x} \]

   4. unpow2 N/A

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

   5. lower-*.f64 27.5

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

8. Applied rewrites 27.5%

   \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    2. metadata-eval N/A

       \[\leadsto \left(\left(x + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    3. sub-neg N/A

       \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    4. lower--.f64 29.4

       \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

11. Applied rewrites 29.4%

    \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]
12. Add Preprocessing

Alternative 3: 24.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod 1.0 (fma (* x x) -0.25 1.0)) (exp (- x))))

C

double code(double x) {
	return fmod(1.0, fma((x * x), -0.25, 1.0)) * exp(-x);
}

Julia

function code(x)
	return Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
4. Step-by-step derivation

5. Applied rewrites 27.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + 1\right)\right) \cdot e^{-x} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, 1\right)\right)}\right) \cdot e^{-x} \]

   4. unpow2 N/A

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

   5. lower-*.f64 27.5

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

8. Applied rewrites 27.5%

   \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]
9. Add Preprocessing

Alternative 4: 22.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\left(x \cdot x\right) \cdot -0.25\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod 1.0 (* (* x x) -0.25)) (exp (- x))))

C

double code(double x) {
	return fmod(1.0, ((x * x) * -0.25)) * exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(1.0d0, ((x * x) * (-0.25d0))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(1.0, ((x * x) * -0.25)) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
4. Step-by-step derivation

5. Applied rewrites 27.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + 1\right)\right) \cdot e^{-x} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, 1\right)\right)}\right) \cdot e^{-x} \]

   4. unpow2 N/A

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

   5. lower-*.f64 27.5

      \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

8. Applied rewrites 27.5%

   \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

9. Taylor expanded in x around inf

   \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot e^{-x} \]
10. Step-by-step derivation

11. Applied rewrites 26.7%

    \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{-0.25}\right)\right) \cdot e^{-x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))