expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 26.6%

Time: 10.4s

Alternatives: 6

Speedup: 3.9×

• could not determine a ground truth

  1. x = -2.3467312623486247e+185

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: 26.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{1}{e^{x}} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 2.0)
   (*
    (/ 1.0 (exp x))
    (fmod
     (exp x)
     (fma
      (fma
       (fma -0.003298611111111111 (* x x) -0.010416666666666666)
       (* x x)
       -0.25)
      (* x x)
      1.0)))
   (* 1.0 (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = (1.0 / exp(x)) * fmod(exp(x), fma(fma(fma(-0.003298611111111111, (x * x), -0.010416666666666666), (x * x), -0.25), (x * x), 1.0));
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(Float64(1.0 / exp(x)) * rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}, {x}^{2}, \frac{-1}{4}\right)}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot {x}^{2} + \color{blue}{\frac{-1}{96}}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-19}{5760}, {x}^{2}, \frac{-1}{96}\right)}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, \color{blue}{x \cdot x}, \frac{-1}{96}\right), {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, \color{blue}{x \cdot x}, \frac{-1}{96}\right), {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      13. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      16. lower-*.f64 9.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{e^{-x}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      3. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \frac{1}{\color{blue}{e^{x}}} \]

      5. lower-/.f64 9.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

   7. Applied rewrites 9.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

   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 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{1}{e^{x}} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 26.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (*
      (fmod
       (exp x)
       (fma
        (fma
         (fma -0.003298611111111111 (* x x) -0.010416666666666666)
         (* x x)
         -0.25)
        (* x x)
        1.0))
      t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), fma(fma(fma(-0.003298611111111111, (x * x), -0.010416666666666666), (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2}} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, {x}^{2}, 1\right)\right)}\right) \cdot e^{-x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}, {x}^{2}, \frac{-1}{4}\right)}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot {x}^{2} + \color{blue}{\frac{-1}{96}}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-19}{5760}, {x}^{2}, \frac{-1}{96}\right)}, {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, \color{blue}{x \cdot x}, \frac{-1}{96}\right), {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, \color{blue}{x \cdot x}, \frac{-1}{96}\right), {x}^{2}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      13. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

      16. lower-*.f64 9.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), \color{blue}{x \cdot x}, 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   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 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 26.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\left(\frac{1}{t\_0}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 x -1.0) x 1.0)))
   (if (<= x 0.01)
     (* (fmod (/ 1.0 t_0) (sqrt (cos x))) t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fma(fma(0.5, x, -1.0), x, 1.0);
	double tmp;
	if (x <= 0.01) {
		tmp = fmod((1.0 / t_0), sqrt(cos(x))) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.5, x, -1.0), x, 1.0)
	tmp = 0.0
	if (x <= 0.01)
		tmp = Float64(rem(Float64(1.0 / t_0), sqrt(cos(x))) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.01], N[(N[With[{TMP1 = N[(1.0 / t$95$0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0100000000000000002

   1. Initial program 9.4%

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

      1. remove-double-div N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 9.4

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

   4. Applied rewrites 9.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 8.2

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

   7. Applied rewrites 8.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(\left(\frac{1}{\color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, 1\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{x \cdot \left(\frac{1}{2} \cdot x - 1\right) + 1}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\left(\frac{1}{2} \cdot x - 1\right) \cdot x} + 1}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, 1\right) \]

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot x + \color{blue}{-1}, x, 1\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, 1\right) \]

      6. lower-fma.f64 9.4

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

   10. Applied rewrites 9.4%

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

   if 0.0100000000000000002 < 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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 24.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 - 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 7.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. sub-neg N/A

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

    4. lower--.f64 23.5

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

11. Applied rewrites 23.5%

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

12. Taylor expanded in x around 0

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

    1. neg-mul-1 N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 23.9

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

14. Applied rewrites 23.9%

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

15. Final simplification 23.9%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(x - -1\right) \bmod 1\right) \]
16. Add Preprocessing

Alternative 5: 24.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. sub-neg N/A

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

    4. lower--.f64 23.5

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

11. Applied rewrites 23.5%

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

Alternative 6: 23.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 22.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 22.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 22.0%

    \[\leadsto \left(1 \bmod 1\right) \cdot \color{blue}{1} \]

12. Final simplification 22.0%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))