expfmod (used to be hard to sample)

Average Error: 59.6 → 24.0

Time: 14.1s

Precision: binary64

Cost: 251656

Math

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

↓

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := \frac{t_0}{e^{x}}\\ t_3 := \sqrt[3]{t_2}\\ t_4 := {t_3}^{2}\\ t_5 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t_4}, {\left(\sqrt[3]{t_3}\right)}^{4}, t_2 - t_4\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (* t_0 (exp (- x))))
        (t_2 (/ t_0 (exp x)))
        (t_3 (cbrt t_2))
        (t_4 (pow t_3 2.0))
        (t_5 (pow (+ (expm1 (* x -0.3333333333333333)) 1.0) 3.0)))
   (if (<= t_1 0.0)
     t_5
     (if (<= t_1 2.0) (fma (cbrt t_4) (pow (cbrt t_3) 4.0) (- t_2 t_4)) t_5))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = t_0 * exp(-x);
	double t_2 = t_0 / exp(x);
	double t_3 = cbrt(t_2);
	double t_4 = pow(t_3, 2.0);
	double t_5 = pow((expm1((x * -0.3333333333333333)) + 1.0), 3.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_5;
	} else if (t_1 <= 2.0) {
		tmp = fma(cbrt(t_4), pow(cbrt(t_3), 4.0), (t_2 - t_4));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

↓

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = Float64(t_0 * exp(Float64(-x)))
	t_2 = Float64(t_0 / exp(x))
	t_3 = cbrt(t_2)
	t_4 = t_3 ^ 2.0
	t_5 = Float64(expm1(Float64(x * -0.3333333333333333)) + 1.0) ^ 3.0
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_5;
	elseif (t_1 <= 2.0)
		tmp = fma(cbrt(t_4), (cbrt(t_3) ^ 4.0), Float64(t_2 - t_4));
	else
		tmp = t_5;
	end
	return tmp
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]

↓

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[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(Exp[N[(x * -0.3333333333333333), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$5, If[LessEqual[t$95$1, 2.0], N[(N[Power[t$95$4, 1/3], $MachinePrecision] * N[Power[N[Power[t$95$3, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[(t$95$2 - t$95$4), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]

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))) < 0.0 or 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 61.8

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

   2. Simplified 61.8

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      Proof
      (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) 1)) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (/.f64 1 (exp.f64 x)))): 1 points increase in error, 1 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 61.8

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{3}} \]

   4. Applied egg-rr 61.8

      \[\leadsto {\color{blue}{\left(\left(1 + \sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right) - 1\right)}}^{3} \]

   5. Applied egg-rr 61.8

      \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right) + 1\right)}}^{3} \]

   6. Taylor expanded in x around inf 24.5

      \[\leadsto {\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) + 1\right)}^{3} \]

   7. Simplified 24.5

      \[\leadsto {\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \color{blue}{\left(-x\right)}\right) + 1\right)}^{3} \]
      Proof
      (neg.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)): 0 points increase in error, 0 points decrease in error

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9

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

   2. Simplified 12.7

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      Proof
      (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) 1)) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (/.f64 1 (exp.f64 x)))): 1 points increase in error, 1 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 12.7

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{3}} \]

   4. Applied egg-rr 12.8

      \[\leadsto {\color{blue}{\left(\left(1 + \sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right) - 1\right)}}^{3} \]

   5. Applied egg-rr 12.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}}, \sqrt[3]{\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}} \cdot \sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}, \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right) \cdot {\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}\right)} \]

   6. Simplified 12.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}}, {\left(\sqrt[3]{\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)}^{4}, \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - {\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}\right)} \]
      Proof
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (pow.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) 4) (-.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (pow.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (Rewrite<= metadata-eval (+.f64 3 1))) (-.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) 3) (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))))) (-.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (Rewrite=> rem-cube-cbrt_binary64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))))) (-.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))): 2 points increase in error, 1 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (Rewrite<= *-commutative_binary64 (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))))) (-.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (neg.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (+.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (+.f64 (Rewrite<= rem-3cbrt-rft_binary64 (*.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) (*.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))))) (*.f64 -1 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (+.f64 (*.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) (Rewrite<= unpow2_binary64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2))) (*.f64 -1 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (+.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) -1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (+.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (Rewrite<= sub-neg_binary64 (-.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (Rewrite<= unpow1/3_binary64 (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) 1/3)) 1))): 1 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (Rewrite<= metadata-eval (*.f64 2 1/6))) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) 1/6) (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) 1/6))) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (*.f64 2 1/6))) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (pow.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)) (Rewrite=> metadata-eval 1/3)) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (pow.f64 (/.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))))) (exp.f64 x)) 1/3) 1))): 1 points increase in error, 1 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (pow.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))) x))) 1/3) 1))): 1 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (-.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))) x) 1/3))) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (-.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/3 (-.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))) x)))) 1))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (*.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2) (Rewrite=> expm1-def_binary64 (expm1.f64 (*.f64 1/3 (-.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))) x)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cbrt.f64 (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)) (*.f64 (cbrt.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)))) (Rewrite<= *-commutative_binary64 (*.f64 (expm1.f64 (*.f64 1/3 (-.f64 (log.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x)))) x))) (pow.f64 (cbrt.f64 (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x))) 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 24.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;{\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}}, {\left(\sqrt[3]{\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)}^{4}, \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - {\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.0
Cost	122696

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;{\left({\left(\sqrt[3]{e}\right)}^{\left(\log t_0 - x\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	24.0
Cost	110152

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;{\left(\left(1 + \sqrt[3]{\frac{t_0}{e^{x}}}\right) + -1\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	24.0
Cost	109896

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;{\left(\sqrt[3]{\frac{t_0}{e^{x}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	24.0
Cost	97160

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	24.0
Cost	97032

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t_0 \cdot e^{-x}\\ t_2 := {\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\frac{t_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	24.7
Cost	19588

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3}\\ \end{array} \]

Alternative 7
Error	25.3
Cost	13184

\[{\left(\mathsf{expm1}\left(x \cdot -0.3333333333333333\right) + 1\right)}^{3} \]

Alternative 8
Error	60.5
Cost	12928

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

Reproduce

herbie shell --seed 2022339 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))