expfmod (used to be hard to sample)

Average Error: 93.2% → 93.21%

Time: 16.6s

Precision: binary64

Cost: 161856

Math

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

↓

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

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)))
   (/
    (* (sqrt t_0) (sqrt (+ -1.0 (pow t_1 3.0))))
    (sqrt (+ (pow t_1 2.0) (+ t_0 2.0))))))

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;
	return (sqrt(t_0) * sqrt((-1.0 + pow(t_1, 3.0)))) / sqrt((pow(t_1, 2.0) + (t_0 + 2.0)));
}

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
    t_0 = mod(exp(x), sqrt(cos(x))) / exp(x)
    t_1 = t_0 + 1.0d0
    code = (sqrt(t_0) * sqrt(((-1.0d0) + (t_1 ** 3.0d0)))) / sqrt(((t_1 ** 2.0d0) + (t_0 + 2.0d0)))
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
	return (math.sqrt(t_0) * math.sqrt((-1.0 + math.pow(t_1, 3.0)))) / math.sqrt((math.pow(t_1, 2.0) + (t_0 + 2.0)))

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)
	return Float64(Float64(sqrt(t_0) * sqrt(Float64(-1.0 + (t_1 ^ 3.0)))) / sqrt(Float64((t_1 ^ 2.0) + Float64(t_0 + 2.0))))
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]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(-1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 93.2

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

2. Simplified 93.19

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

   [Start] 93.2	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   exp-neg [=>] 93.2	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
   associate-*r/ [=>] 93.19	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
   *-rgt-identity [=>] 93.19	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

3. Applied egg-rr 93.21

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

4. Applied egg-rr 93.21

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

5. Final simplification 93.21

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

Alternatives

Alternative 1
Error	93.21%
Cost	45056

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

Alternative 2
Error	93.18%
Cost	38656

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

Alternative 3
Error	93.19%
Cost	32256

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

Alternative 4
Error	93.39%
Cost	19840

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

Alternative 5
Error	93.63%
Cost	19456

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

Alternative 6
Error	94.06%
Cost	13568

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

Alternative 7
Error	93.95%
Cost	13568

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

Alternative 8
Error	94.07%
Cost	13440

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

Alternative 9
Error	94.07%
Cost	13184

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

Alternative 10
Error	94.53%
Cost	12928

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

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))