expfmod (used to be hard to sample)

Percentage Accurate: 6.3% → 99.0%

Time: 10.8s

Alternatives: 5

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.8296387205384917e+48

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: 6.3% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ t_2 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{\left({\left(e^{-1}\right)}^{\left(x \cdot x - {t\_0}^{2}\right)}\right)}^{\left({\left(t\_0 + x\right)}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (fma (* -0.25 x) x 1.0))))
        (t_1 (fma (* x x) -0.25 1.0))
        (t_2 (exp (- x))))
   (if (<= x -2e-310)
     (pow (pow (exp -1.0) (- (* x x) (pow t_0 2.0))) (pow (+ t_0 x) -1.0))
     (if (<= x 0.5)
       (* (fmod (* (fma 0.5 x 1.0) x) t_1) t_2)
       (* (fmod 1.0 t_1) t_2)))))

C

double code(double x) {
	double t_0 = log(fmod(exp(x), fma((-0.25 * x), x, 1.0)));
	double t_1 = fma((x * x), -0.25, 1.0);
	double t_2 = exp(-x);
	double tmp;
	if (x <= -2e-310) {
		tmp = pow(pow(exp(-1.0), ((x * x) - pow(t_0, 2.0))), pow((t_0 + x), -1.0));
	} else if (x <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), t_1) * t_2;
	} else {
		tmp = fmod(1.0, t_1) * t_2;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = log(rem(exp(x), fma(Float64(-0.25 * x), x, 1.0)))
	t_1 = fma(Float64(x * x), -0.25, 1.0)
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-310)
		tmp = (exp(-1.0) ^ Float64(Float64(x * x) - (t_0 ^ 2.0))) ^ (Float64(t_0 + x) ^ -1.0);
	elseif (x <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), t_1) * t_2);
	else
		tmp = Float64(rem(1.0, t_1) * t_2);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-310], N[Power[N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$0 + x), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 6.0

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

   5. Applied rewrites 6.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 6.0

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

   7. Applied rewrites 6.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 6.0

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

   9. Applied rewrites 6.0%

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

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{{\left({\left(e^{-1}\right)}^{\left(x \cdot x - {\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)}^{2}\right)}\right)}^{\left({\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right) + x\right)}^{-1}\right)}} \]

   if -1.999999999999994e-310 < x < 0.5

   1. Initial program 10.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 5.9

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

   8. Applied rewrites 5.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 9.7

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

   11. Applied rewrites 9.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 96.9%

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

   if 0.5 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 61.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_1) 0.5)
     (* (fmod (* (fma 0.5 x 1.0) x) t_0) t_1)
     (* (fmod (+ 1.0 x) t_0) t_1))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_1) <= 0.5) {
		tmp = fmod((fma(0.5, x, 1.0) * x), t_0) * t_1;
	} else {
		tmp = fmod((1.0 + x), t_0) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_1) <= 0.5)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), t_0) * t_1);
	else
		tmp = Float64(rem(Float64(1.0 + x), t_0) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision], 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $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))) < 0.5

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.5

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

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 6.3

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

   11. Applied rewrites 6.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 48.6%

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

   if 0.5 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 95.6

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 98.8

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

   11. Applied rewrites 98.8%

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

Alternative 3: 24.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (fma (* x x) -0.25 1.0)))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (fmod (exp x) t_1)
     (* (fmod 1.0 t_1) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fma((x * x), -0.25, 1.0);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), t_1);
	} else {
		tmp = fmod(1.0, t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = rem(exp(x), t_1);
	else
		tmp = Float64(rem(1.0, t_1) * t_0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
   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 6.7

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 4: 24.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 26.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 26.6

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

8. Applied rewrites 26.6%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 28.5

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

11. Applied rewrites 28.5%

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

Alternative 5: 5.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around 0

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

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

8. Applied rewrites 5.2%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))