expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 6.9%

Time: 14.3s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = -6.347605038217588e+239

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: 6.9% 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: 6.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ (fmod (exp x) (sqrt (cos x))) (exp x)))))
   (fma (+ t_0 1.0) (+ t_0 -1.0) 1.0)))

C

double code(double x) {
	double t_0 = sqrt((fmod(exp(x), sqrt(cos(x))) / exp(x)));
	return fma((t_0 + 1.0), (t_0 + -1.0), 1.0);
}

Julia

function code(x)
	t_0 = sqrt(Float64(rem(exp(x), sqrt(cos(x))) / exp(x)))
	return fma(Float64(t_0 + 1.0), Float64(t_0 + -1.0), 1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[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]], $MachinePrecision]}, N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

   1. add-cbrt-cube 6.8%

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

   2. pow1/3 6.8%

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

   3. add-sqr-sqrt 6.8%

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

   4. pow1 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\color{blue}{{\cos x}^{1}} \cdot \sqrt{\cos x}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   5. pow1/2 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{1} \cdot \color{blue}{{\cos x}^{0.5}}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   6. pow-prod-up 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\color{blue}{\left({\cos x}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   7. metadata-eval 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

5. Applied egg-rr 6.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{1.5}\right)}^{0.3333333333333333}\right)}\right)}{e^{x}} \]
6. Step-by-step derivation

   1. unpow1/3 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{{\cos x}^{1.5}}\right)}\right)}{e^{x}} \]

7. Simplified 6.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{{\cos x}^{1.5}}\right)}\right)}{e^{x}} \]
8. Step-by-step derivation

   1. pow1/3 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{1.5}\right)}^{0.3333333333333333}\right)}\right)}{e^{x}} \]

   2. pow-pow 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}\right)}{e^{x}} \]

   3. metadata-eval 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\cos x}^{\color{blue}{0.5}}\right)\right)}{e^{x}} \]

   4. pow1/2 6.8%

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

   5. expm1-log1p-u 6.8%

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

   6. expm1-udef 6.8%

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

   7. log1p-udef 6.8%

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

   8. add-exp-log 6.8%

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

   9. associate--l+ 6.8%

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

   10. +-commutative 6.8%

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

   11. add-sqr-sqrt 6.8%

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

   12. difference-of-sqr-1 6.8%

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

9. Applied egg-rr 6.8%

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

10. Final simplification 6.8%

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

Alternative 2: 6.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (cbrt (pow (cos x) 1.5))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), cbrt(pow(cos(x), 1.5))) / exp(x);
}

Julia

function code(x)
	return Float64(rem(exp(x), cbrt((cos(x) ^ 1.5))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[Power[N[Cos[x], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

   1. add-cbrt-cube 6.8%

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

   2. pow1/3 6.8%

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

   3. add-sqr-sqrt 6.8%

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

   4. pow1 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\color{blue}{{\cos x}^{1}} \cdot \sqrt{\cos x}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   5. pow1/2 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{1} \cdot \color{blue}{{\cos x}^{0.5}}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   6. pow-prod-up 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\color{blue}{\left({\cos x}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)\right)}{e^{x}} \]

   7. metadata-eval 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left({\cos x}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)\right)}{e^{x}} \]

5. Applied egg-rr 6.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({\left({\cos x}^{1.5}\right)}^{0.3333333333333333}\right)}\right)}{e^{x}} \]
6. Step-by-step derivation

   1. unpow1/3 6.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{{\cos x}^{1.5}}\right)}\right)}{e^{x}} \]

7. Simplified 6.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{{\cos x}^{1.5}}\right)}\right)}{e^{x}} \]

8. Final simplification 6.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{e^{x}} \]

Alternative 3: 6.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ (/ (fmod (exp x) (sqrt (cos x))) (exp x)) 1.0) -1.0))

C

double code(double x) {
	return ((fmod(exp(x), sqrt(cos(x))) / exp(x)) + 1.0) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((mod(exp(x), sqrt(cos(x))) / exp(x)) + 1.0d0) + (-1.0d0)
end function

Python

def code(x):
	return ((math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x)) + 1.0) + -1.0

Julia

function code(x)
	return Float64(Float64(Float64(rem(exp(x), sqrt(cos(x))) / exp(x)) + 1.0) + -1.0)
end

Wolfram

code[x_] := N[(N[(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] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

   1. expm1-log1p-u 6.8%

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

   2. expm1-udef 6.8%

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

   3. log1p-udef 6.8%

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

   4. add-exp-log 6.8%

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

5. Applied egg-rr 6.8%

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

6. Final simplification 6.8%

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

Alternative 4: 6.9% accurate, 1.0× speedup

Math

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

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(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]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

4. Final simplification 6.8%

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

Alternative 5: 6.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fmod (exp x) 1.0) (exp x)))

C

double code(double x) {
	return fmod(exp(x), 1.0) / exp(x);
}

Fortran

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

Python

def code(x):
	return math.fmod(math.exp(x), 1.0) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

4. Taylor expanded in x around 0 6.6%

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

5. Final simplification 6.6%

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

Alternative 6: 5.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1 + x \cdot x}{\frac{x + 1}{\left(\left(e^{x}\right) \bmod 1\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 1.0 (* x x)) (/ (+ x 1.0) (fmod (exp x) 1.0))))

C

double code(double x) {
	return (1.0 + (x * x)) / ((x + 1.0) / fmod(exp(x), 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + (x * x)) / ((x + 1.0d0) / mod(exp(x), 1.0d0))
end function

Python

def code(x):
	return (1.0 + (x * x)) / ((x + 1.0) / math.fmod(math.exp(x), 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(x * x)) / Float64(Float64(x + 1.0) / rem(exp(x), 1.0)))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

4. Taylor expanded in x around 0 6.6%

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

5. Taylor expanded in x around 0 5.7%

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

   1. +-commutative 5.7%

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

   2. *-lft-identity 5.7%

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

   3. associate-*r* 5.7%

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

   4. neg-mul-1 5.7%

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

   5. distribute-rgt-out 5.7%

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

7. Simplified 5.7%

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

   1. flip-+ 5.7%

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

   2. associate-*r/ 5.7%

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

   3. metadata-eval 5.7%

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

   4. sub-neg 5.7%

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

   5. add-sqr-sqrt 3.1%

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

   6. sqrt-unprod 5.8%

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

   7. sqr-neg 5.8%

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

   8. sqrt-unprod 2.6%

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

   9. add-sqr-sqrt 5.8%

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

   10. distribute-rgt-neg-out 5.8%

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

   11. sqr-neg 5.8%

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

   12. neg-sub0 5.8%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 + x \cdot x\right)}{1 - \color{blue}{\left(0 - x\right)}} \]

   13. metadata-eval 5.8%

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

   14. associate--r- 5.8%

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

   15. metadata-eval 5.8%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 + x \cdot x\right)}{\left(1 - \color{blue}{0}\right) + x} \]

   16. metadata-eval 5.8%

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

9. Applied egg-rr 5.8%

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

    1. *-commutative 5.8%

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

    2. associate-/l* 5.8%

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

    3. +-commutative 5.8%

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

11. Simplified 5.8%

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

12. Final simplification 5.8%

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

Alternative 7: 5.9% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) 1.0) (- 1.0 x)))

C

double code(double x) {
	return fmod(exp(x), 1.0) * (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), 1.0d0) * (1.0d0 - x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), 1.0) * (1.0 - x)

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) * Float64(1.0 - x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

4. Taylor expanded in x around 0 6.6%

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

5. Taylor expanded in x around 0 5.7%

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

   1. +-commutative 5.7%

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

   2. *-lft-identity 5.7%

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

   3. associate-*r* 5.7%

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

   4. neg-mul-1 5.7%

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

   5. distribute-rgt-out 5.7%

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

7. Simplified 5.7%

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

8. Taylor expanded in x around 0 5.7%

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

   1. +-commutative 5.7%

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

   2. mul-1-neg 5.7%

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

   3. *-commutative 5.7%

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

   4. sub-neg 5.7%

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

   5. *-rgt-identity 5.7%

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

   6. distribute-lft-out-- 5.7%

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

10. Simplified 5.7%

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

11. Final simplification 5.7%

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

Alternative 8: 5.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod (exp x) 1.0))

C

double code(double x) {
	return fmod(exp(x), 1.0);
}

Fortran

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

Python

def code(x):
	return math.fmod(math.exp(x), 1.0)

Julia

function code(x)
	return rem(exp(x), 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. associate-*r/ 6.8%

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

   3. *-rgt-identity 6.8%

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

3. Simplified 6.8%

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

4. Taylor expanded in x around 0 6.6%

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

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

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

Reproduce

herbie shell --seed 2023293 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))