expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 62.6%

Time: 21.8s

Alternatives: 4

Speedup: 505.0×

• could not determine a ground truth

  1. x = -1.236030892974936e+275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Alternative 1: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\cos x}}\\ t_1 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({t\_0}^{2}\right) + \log t\_0}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (cos x)))) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_1) 2.0)
     (/ (fmod (exp x) (sqrt (+ (log (pow t_0 2.0)) (log t_0)))) (exp x))
     t_1)))

C

double code(double x) {
	double t_0 = cbrt(exp(cos(x)));
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_1) <= 2.0) {
		tmp = fmod(exp(x), sqrt((log(pow(t_0, 2.0)) + log(t_0)))) / exp(x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], t$95$1]]]

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))) < 2

   1. Initial program 10.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. /-rgt-identity 10.5%

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

      2. associate-/r/ 10.5%

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

      3. exp-neg 10.5%

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

      4. remove-double-neg 10.5%

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

   3. Simplified 10.5%

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

      1. add-log-exp 10.5%

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

      2. add-cube-cbrt 56.5%

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

      3. log-prod 56.5%

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

      4. pow2 56.5%

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

   6. Applied egg-rr 56.5%

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

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

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. exp-neg 0.0%

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

      2. div-inv 0.0%

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

      3. add-cbrt-cube 0.0%

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

      4. pow1/3 0.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

      5. pow-to-exp 0.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot 0.3333333333333333}} \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in x around inf 1.6%

      \[\leadsto e^{\color{blue}{\left(3 \cdot x\right)} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. *-commutative 1.6%

         \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

   7. Simplified 1.6%

      \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

      1. associate-*l* 1.6%

         \[\leadsto e^{\color{blue}{x \cdot \left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 1.6%

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

      3. pow-exp 1.6%

         \[\leadsto \color{blue}{{\left(e^{x}\right)}^{1}} \]

      4. pow1 1.6%

         \[\leadsto \color{blue}{e^{x}} \]

      5. add-sqr-sqrt 1.6%

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

      6. sqrt-unprod 1.6%

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

      7. sqr-neg 1.6%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. exp-neg 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{x}}} \]
   10. Step-by-step derivation

       1. rec-exp 100.0%

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

   11. Simplified 100.0%

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

Alternative 2: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310) 1.0 (/ (fmod (+ x 1.0) (sqrt (cos x))) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0;
	} else {
		tmp = fmod((x + 1.0), sqrt(cos(x))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = 1.0d0
    else
        tmp = mod((x + 1.0d0), sqrt(cos(x))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = 1.0
	else:
		tmp = math.fmod((x + 1.0), math.sqrt(math.cos(x))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = 1.0;
	else
		tmp = Float64(rem(Float64(x + 1.0), sqrt(cos(x))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], 1.0, N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 9.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. exp-neg 9.5%

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

      2. div-inv 9.5%

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

      3. add-cbrt-cube 9.5%

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

      4. pow1/3 9.5%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

      5. pow-to-exp 9.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot 0.3333333333333333}} \]

   4. Applied egg-rr 6.9%

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

   5. Taylor expanded in x around inf 97.5%

      \[\leadsto e^{\color{blue}{\left(3 \cdot x\right)} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. *-commutative 97.5%

         \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

   7. Simplified 97.5%

      \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

   8. Taylor expanded in x around 0 100.0%

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

   if -3.999999999999988e-310 < x

   1. Initial program 7.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. /-rgt-identity 7.9%

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

      2. associate-/r/ 7.9%

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

      3. exp-neg 7.9%

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

      4. remove-double-neg 7.9%

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

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 38.4%

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

      1. +-commutative 38.4%

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

   7. Simplified 38.4%

      \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right)} \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

code[x_] := N[Exp[(-x)], $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. exp-neg 8.6%

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

   2. div-inv 8.6%

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

   3. add-cbrt-cube 8.6%

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

   4. pow1/3 8.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

   5. pow-to-exp 8.6%

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot 0.3333333333333333}} \]

4. Applied egg-rr 6.4%

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

5. Taylor expanded in x around inf 43.3%

   \[\leadsto e^{\color{blue}{\left(3 \cdot x\right)} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. *-commutative 43.3%

      \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

7. Simplified 43.3%

   \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]
8. Step-by-step derivation

   1. associate-*l* 43.3%

      \[\leadsto e^{\color{blue}{x \cdot \left(3 \cdot 0.3333333333333333\right)}} \]

   2. metadata-eval 43.3%

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

   3. pow-exp 43.3%

      \[\leadsto \color{blue}{{\left(e^{x}\right)}^{1}} \]

   4. pow1 43.3%

      \[\leadsto \color{blue}{e^{x}} \]

   5. add-sqr-sqrt 2.5%

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

   6. sqrt-unprod 43.3%

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

   7. sqr-neg 43.3%

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

   8. sqrt-unprod 40.8%

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

   9. add-sqr-sqrt 61.8%

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

   10. exp-neg 61.8%

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

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\frac{1}{e^{x}}} \]
10. Step-by-step derivation

    1. rec-exp 61.8%

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

11. Simplified 61.8%

    \[\leadsto \color{blue}{e^{-x}} \]
12. Add Preprocessing

Alternative 4: 43.3% accurate, 505.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 8.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. exp-neg 8.6%

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

   2. div-inv 8.6%

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

   3. add-cbrt-cube 8.6%

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

   4. pow1/3 8.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

   5. pow-to-exp 8.6%

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot 0.3333333333333333}} \]

4. Applied egg-rr 6.4%

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

5. Taylor expanded in x around inf 43.3%

   \[\leadsto e^{\color{blue}{\left(3 \cdot x\right)} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. *-commutative 43.3%

      \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

7. Simplified 43.3%

   \[\leadsto e^{\color{blue}{\left(x \cdot 3\right)} \cdot 0.3333333333333333} \]

8. Taylor expanded in x around 0 44.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))