expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 60.7%

Time: 10.8s

Alternatives: 6

Speedup: 4.1×

• could not determine a ground truth

  1. x = -4.4056176622470364e+306

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: 6.7% 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: 60.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.1:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.1)
   (/ (fmod (fma x (* x 0.5) x) (fma -0.25 (* x x) 1.0)) (exp x))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.1) {
		tmp = fmod(fma(x, (x * 0.5), x), fma(-0.25, (x * x), 1.0)) / exp(x);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.1)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), fma(-0.25, Float64(x * x), 1.0)) / exp(x));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := 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] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.1], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 0.10000000000000001

   1. Initial program 6.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.5

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.5%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-exp.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. exp-neg N/A

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

       5. un-div-inv N/A

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

       6. lower-/.f64 N/A

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

   13. Applied rewrites 47.4%

       \[\leadsto \color{blue}{\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}} \]

   if 0.10000000000000001 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   8. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 97.9

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

   10. Applied rewrites 97.9%

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

Alternative 2: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0.1:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.1)
     (* t_0 (fmod (fma x (* x 0.5) x) 1.0))
     (/ (fmod (+ x 1.0) 1.0) (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.1) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.1)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	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], 0.1], N[(t$95$0 * N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 0.10000000000000001

   1. Initial program 6.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.3

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod 1\right) \cdot e^{-x} \]

   8. Applied rewrites 5.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod 1\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod 1\right) \cdot e^{-x} \]

   if 0.10000000000000001 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   8. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 97.9

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

   10. Applied rewrites 97.9%

       \[\leadsto \color{blue}{\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.1:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 25.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fmod (+ x 1.0) 1.0) (exp x)))

C

double code(double x) {
	return fmod((x + 1.0), 1.0) / exp(x);
}

Fortran

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

Python

def code(x):
	return math.fmod((x + 1.0), 1.0) / math.exp(x)

Julia

function code(x)
	return Float64(rem(Float64(x + 1.0), 1.0) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 24.8

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

8. Applied rewrites 24.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 24.8

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

10. Applied rewrites 24.8%

    \[\leadsto \color{blue}{\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}} \]
11. Add Preprocessing

Alternative 4: 25.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 24.8

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

8. Applied rewrites 24.8%

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

9. Final simplification 24.8%

   \[\leadsto e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right) \]
10. Add Preprocessing

Alternative 5: 24.2% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fmod (+ x 1.0) 1.0))

C

double code(double x) {
	return fmod((x + 1.0), 1.0);
}

Fortran

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

Python

def code(x):
	return math.fmod((x + 1.0), 1.0)

Julia

function code(x)
	return rem(Float64(x + 1.0), 1.0)
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.2

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

5. Applied rewrites 5.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(1 + x\right) \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 23.7%

    \[\leadsto \left(\left(x + 1\right) \bmod 1\right) \]
12. Add Preprocessing

Alternative 6: 23.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod 1.0 1.0))

C

double code(double x) {
	return fmod(1.0, 1.0);
}

Fortran

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

Python

def code(x):
	return math.fmod(1.0, 1.0)

Julia

function code(x)
	return rem(1.0, 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.2

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

5. Applied rewrites 5.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 22.6%

    \[\leadsto \left(1 \bmod 1\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))