expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 62.6%

Time: 13.0s

Alternatives: 6

Speedup: 3.7×

• could not determine a ground truth

  1. x = -5.14146459550679e+254

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: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 10^{-13}:\\ \;\;\;\;t\_1 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 1e-13)
     (* t_1 (fmod (fma x (* x 0.5) x) 1.0))
     (if (<= t_2 2.0) (/ t_0 (exp x)) (/ (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 1e-13) {
		tmp = t_1 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else if (t_2 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 1e-13)
		tmp = Float64(t_1 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{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]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-13], N[(t$95$1 * N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1e-13

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 4.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 48.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 48.9%

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

   if 1e-13 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   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

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 10^{-13}:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 1e-13)
     (* t_0 (fmod (fma x (* x 0.5) x) 1.0))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (fma
          (* x x)
          (fma
           x
           (* x (fma (* x x) -0.003298611111111111 -0.010416666666666666))
           -0.25)
          1.0)))
       (/ (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma(x, (x * fma((x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0)));
	else
		tmp = Float64(rem(1.0, 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e-13], 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], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1e-13

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 4.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 48.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 48.9%

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

   if 1e-13 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   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

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 10^{-13}:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0.02:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.02)
     (* t_0 (fmod (fma x (* x 0.5) x) 1.0))
     (* t_0 (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.02) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = t_0 * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.02)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	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], 0.02], 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[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], 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))) < 0.0200000000000000004

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.1

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 6.1%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 48.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 48.6%

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

   if 0.0200000000000000004 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 7.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 7.2%

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

   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 94.2

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

   8. Applied rewrites 94.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.02:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.9% 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 7.1%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0%

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

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 24.7

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

8. Applied rewrites 24.7%

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

9. Final simplification 24.7%

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

Alternative 5: 24.6% accurate, 3.6× speedup

Math

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

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 7.1%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 6.0

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

7. Applied rewrites 6.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.2%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 23.9

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

13. Applied rewrites 23.9%

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

Alternative 6: 23.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 6.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 6.0

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

7. Applied rewrites 6.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.2%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 22.5%

    \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{1} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))