expfmod (used to be hard to sample)

Average Accuracy: 6.7% → 6.7%

Time: 18.6s

Precision: binary64

Cost: 188288

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ t_1 := t_0 + 1\\ t_2 := {t_1}^{3}\\ \frac{t_2 \cdot t_2 + -1}{\left({t_1}^{2} + \left(t_0 + 2\right)\right) \cdot \left(1 + t_2\right)} \end{array} \]

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))) (exp x)))
        (t_1 (+ t_0 1.0))
        (t_2 (pow t_1 3.0)))
   (/ (+ (* t_2 t_2) -1.0) (* (+ (pow t_1 2.0) (+ t_0 2.0)) (+ 1.0 t_2)))))

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))) / exp(x);
	double t_1 = t_0 + 1.0;
	double t_2 = pow(t_1, 3.0);
	return ((t_2 * t_2) + -1.0) / ((pow(t_1, 2.0) + (t_0 + 2.0)) * (1.0 + t_2));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = mod(exp(x), sqrt(cos(x))) / exp(x)
    t_1 = t_0 + 1.0d0
    t_2 = t_1 ** 3.0d0
    code = ((t_2 * t_2) + (-1.0d0)) / (((t_1 ** 2.0d0) + (t_0 + 2.0d0)) * (1.0d0 + t_2))
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

↓

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x)
	t_1 = t_0 + 1.0
	t_2 = math.pow(t_1, 3.0)
	return ((t_2 * t_2) + -1.0) / ((math.pow(t_1, 2.0) + (t_0 + 2.0)) * (1.0 + t_2))

Julia

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

↓

function code(x)
	t_0 = Float64(rem(exp(x), sqrt(cos(x))) / exp(x))
	t_1 = Float64(t_0 + 1.0)
	t_2 = t_1 ^ 3.0
	return Float64(Float64(Float64(t_2 * t_2) + -1.0) / Float64(Float64((t_1 ^ 2.0) + Float64(t_0 + 2.0)) * Float64(1.0 + t_2)))
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[(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]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 3.0], $MachinePrecision]}, N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.7%

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

2. Simplified 6.7%

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

   [Start] 6.7	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   exp-neg [=>] 6.7	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
   associate-*r/ [=>] 6.7	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
   *-rgt-identity [=>] 6.7	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

3. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1} \]
   Proof

   [Start] 6.7	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
   expm1-log1p-u [=>] 6.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)\right)} \]
   expm1-udef [=>] 6.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1} \]

4. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3} \cdot {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3} - 1}{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{2} + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 2\right)\right) \cdot \left(1 + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3}\right)}} \]
   Proof

   [Start] 6.7	\[ e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1 \]
   flip3-- [=>] 6.7	\[ \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot 1\right)}} \]
   metadata-eval [=>] 6.7	\[ \frac{{\left(e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot 1\right)} \]
   flip-- [=>] 6.7	\[ \frac{\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}\right)}^{3} \cdot {\left(e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}\right)}^{3} - 1 \cdot 1}{{\left(e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}\right)}^{3} + 1}}}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \cdot 1\right)} \]

5. Final simplification 6.7%

   \[\leadsto \frac{{\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3} \cdot {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3} + -1}{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{2} + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 2\right)\right) \cdot \left(1 + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right)}^{3}\right)} \]

Alternatives

Alternative 1
Accuracy	6.7%
Cost	122944

\[\begin{array}{l} t_0 := \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \frac{\sqrt{t_0} \cdot \sqrt{{\left(t_0 + 1\right)}^{2} + -1}}{\sqrt{t_0 + 2}} \end{array} \]

Alternative 2
Accuracy	6.7%
Cost	110336

\[\begin{array}{l} t_0 := \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ t_1 := t_0 + 1\\ \frac{{t_1}^{3} + -1}{{t_1}^{2} + \left(t_0 + 2\right)} \end{array} \]

Alternative 3
Accuracy	6.5%
Cost	45056

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

Alternative 4
Accuracy	6.7%
Cost	32512

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

Alternative 5
Accuracy	6.7%
Cost	32256

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

Alternative 6
Accuracy	6.5%
Cost	26240

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

Alternative 7
Accuracy	6.5%
Cost	19840

\[\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]

Alternative 8
Accuracy	6.3%
Cost	19456

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

Alternative 9
Accuracy	5.7%
Cost	13568

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

Alternative 10
Accuracy	5.7%
Cost	13184

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

Alternative 11
Accuracy	5.3%
Cost	12928

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))