expfmod (used to be hard to sample)

?

Percentage Accurate: 6.7% → 63.6%
Time: 19.8s
Precision: binary64
Cost: 84164

?

\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := e^{-x}\\ t_1 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq 2:\\ \;\;\;\;{\left(e^{{t_1}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (log (fmod (exp x) (sqrt (cos x))))))
   (if (<= x 2.0)
     (pow (exp (- (pow t_1 2.0) (* x x))) (/ 1.0 (+ x t_1)))
     (- t_0 t_0))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = exp(-x);
	double t_1 = log(fmod(exp(x), sqrt(cos(x))));
	double tmp;
	if (x <= 2.0) {
		tmp = pow(exp((pow(t_1, 2.0) - (x * x))), (1.0 / (x + t_1)));
	} else {
		tmp = t_0 - t_0;
	}
	return tmp;
}

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) :: tmp
    t_0 = exp(-x)
    t_1 = log(mod(exp(x), sqrt(cos(x))))
    if (x <= 2.0d0) then
        tmp = exp(((t_1 ** 2.0d0) - (x * x))) ** (1.0d0 / (x + t_1))
    else
        tmp = t_0 - t_0
    end if
    code = tmp
end function

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)
↓
def code(x):
	t_0 = math.exp(-x)
	t_1 = math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x))))
	tmp = 0
	if x <= 2.0:
		tmp = math.pow(math.exp((math.pow(t_1, 2.0) - (x * x))), (1.0 / (x + t_1)))
	else:
		tmp = t_0 - t_0
	return tmp

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end
↓
function code(x)
	t_0 = exp(Float64(-x))
	t_1 = log(rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (x <= 2.0)
		tmp = exp(Float64((t_1 ^ 2.0) - Float64(x * x))) ^ Float64(1.0 / Float64(x + t_1));
	else
		tmp = Float64(t_0 - t_0);
	end
	return tmp
end

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

\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := e^{-x}\\
t_1 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\
\mathbf{if}\;x \leq 2:\\
\;\;\;\;{\left(e^{{t_1}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + t_1}\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0 - t_0\\


\end{array}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 3 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Derivation?

  1. Split input into 2 regimes

  2. if x < 2

    1. Initial program 8.9%

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

    2. Simplified8.9%

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

      [Start]8.9

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

      exp-neg [=>]8.9

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

      associate-*r/ [=>]8.9

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

      *-rgt-identity [=>]8.9

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

    3. Applied egg-rr9.0%

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

      [Start]8.9

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

      add-exp-log [=>]8.9

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

      div-exp [=>]9.0

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

    4. Applied egg-rr53.6%

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

      [Start]9.0

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

      flip-- [=>]5.0

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

      div-inv [=>]5.0

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

      add-log-exp [=>]5.0

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

      exp-to-pow [=>]53.6

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

      pow2 [=>]53.6

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

      +-commutative [=>]53.6

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

    if 2 < x

    1. Initial program 0.0%

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

    2. Simplified0.0%

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

      [Start]0.0

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

      exp-neg [=>]0.0

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

      associate-*r/ [=>]0.0

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

      *-rgt-identity [=>]0.0

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

    3. Applied egg-rr0.0%

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

      [Start]0.0

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

      expm1-log1p-u [=>]0.0

      \[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}}{e^{x}} \]

      expm1-udef [=>]0.0

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

      log1p-udef [=>]0.0

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

      add-exp-log [<=]0.0

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

    4. Applied egg-rr0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - x} + \left(-e^{-x}\right)} \]
      Step-by-step derivation

      [Start]0.0

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

      div-sub [=>]0.0

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

      sub-neg [=>]0.0

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

      add-exp-log [=>]0.0

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

      diff-log [<=]0.0

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

      log1p-udef [<=]0.0

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

      add-log-exp [<=]0.0

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

      rec-exp [=>]0.0

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

    5. Simplified0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - x} - e^{-x}} \]
      Step-by-step derivation

      [Start]0.0

      \[ e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - x} + \left(-e^{-x}\right) \]

      sub-neg [<=]0.0

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

    6. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} - e^{-x} \]

    7. Simplified100.0%

      \[\leadsto e^{\color{blue}{-x}} - e^{-x} \]
      Step-by-step derivation

      [Start]100.0

      \[ e^{-1 \cdot x} - e^{-x} \]

      mul-1-neg [=>]100.0

      \[ e^{\color{blue}{-x}} - e^{-x} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification63.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;{\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x} - e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy62.8%
Cost13252
\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_0\\ \end{array} \]
Alternative 2
Accuracy44.1%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023160 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))