expfmod

Average Error: 59.5 → 24.5

Time: 15.7s

Precision: binary64

Cost: 84360

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\\ \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\frac{1 - {t_0}^{2}}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (expm1 (- (log (fmod (exp x) (sqrt (cos x)))) x))))
   (if (<= x -5.165282355235896e-302)
     1.0
     (if (<= x 3.583668735255346)
       (/ (- 1.0 (pow t_0 2.0)) (- 1.0 t_0))
       (exp (- x))))))

C

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

↓

double code(double x) {
	double t_0 = expm1((log(fmod(exp(x), sqrt(cos(x)))) - x));
	double tmp;
	if (x <= -5.165282355235896e-302) {
		tmp = 1.0;
	} else if (x <= 3.583668735255346) {
		tmp = (1.0 - pow(t_0, 2.0)) / (1.0 - t_0);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.expm1((math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x)))) - x))
	tmp = 0
	if x <= -5.165282355235896e-302:
		tmp = 1.0
	elif x <= 3.583668735255346:
		tmp = (1.0 - math.pow(t_0, 2.0)) / (1.0 - t_0)
	else:
		tmp = math.exp(-x)
	return tmp

Julia

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

↓

function code(x)
	t_0 = expm1(Float64(log(rem(exp(x), sqrt(cos(x)))) - x))
	tmp = 0.0
	if (x <= -5.165282355235896e-302)
		tmp = 1.0;
	elseif (x <= 3.583668735255346)
		tmp = Float64(Float64(1.0 - (t_0 ^ 2.0)) / Float64(1.0 - t_0));
	else
		tmp = exp(Float64(-x));
	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[(Exp[N[(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] - x), $MachinePrecision]] - 1), $MachinePrecision]}, If[LessEqual[x, -5.165282355235896e-302], 1.0, If[LessEqual[x, 3.583668735255346], N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.165282355235896e-302

   1. Initial program 58.1

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

   2. Simplified 58.1

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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)))): 2 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.1

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

   4. Applied egg-rr 58.1

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

   5. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{1} \]

   if -5.165282355235896e-302 < x < 3.5836687352553458

   1. Initial program 58.8

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

   2. Simplified 58.8

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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)))): 2 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.8

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

   4. Applied egg-rr 58.8

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]

   if 3.5836687352553458 < x

   1. Initial program 63.7

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

   2. Simplified 63.7

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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)))): 2 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 63.7

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

   4. Taylor expanded in x around inf 0.5

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

   5. Simplified 0.5

      \[\leadsto e^{\color{blue}{-x}} \]
      Proof
      (neg.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\frac{1 - {\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.5
Cost	71240

\[\begin{array}{l} t_0 := \sqrt[3]{\cos x}\\ \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 118.65353696607265:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left({t_0}^{2}\right)}^{1.5} \cdot {t_0}^{1.5}}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2
Error	24.5
Cost	64584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;e^{{\left(\sqrt[3]{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 3
Error	24.5
Cost	39048

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;1 + \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 4
Error	24.5
Cost	38984

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 118.65353696607265:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 5
Error	24.5
Cost	38920

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 6
Error	24.5
Cost	32520

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 7
Error	24.6
Cost	26888

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 8
Error	24.7
Cost	26376

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 9
Error	24.7
Cost	26248

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 10
Error	24.7
Cost	20104

\[\begin{array}{l} \mathbf{if}\;x \leq -5.165282355235896 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.583668735255346:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 11
Error	25.6
Cost	6528

\[e^{-x} \]

Alternative 12
Error	37.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))