expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 10.6%

Time: 24.0s

Alternatives: 6

Speedup: 1.2×

• could not determine a ground truth

  1. x = -1.8375847468729098e+163

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: 10.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left|\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) + -1\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (fabs (+ (fmod (exp x) (fma x (* x -0.25) 1.0)) -1.0))))

C

double code(double x) {
	return 1.0 + fabs((fmod(exp(x), fma(x, (x * -0.25), 1.0)) + -1.0));
}

Julia

function code(x)
	return Float64(1.0 + abs(Float64(rem(exp(x), fma(x, Float64(x * -0.25), 1.0)) + -1.0)))
end

Wolfram

code[x_] := N[(1.0 + N[Abs[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

   1. expm1-log1p-u 6.3%

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

   2. expm1-udef 6.3%

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

   3. log1p-udef 6.3%

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

   4. add-exp-log 6.3%

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

5. Applied egg-rr 6.3%

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

   1. associate--l+ 6.3%

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

7. Simplified 6.3%

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

8. Taylor expanded in x around 0 5.8%

   \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} - 1\right) \]
9. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} - 1\right) \]

   2. unpow2 5.8%

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

10. Simplified 5.8%

    \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.25\right)}\right)}{e^{x}} - 1\right) \]
11. Step-by-step derivation

    1. add-sqr-sqrt 2.0%

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

    2. sqrt-unprod 10.5%

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

    3. pow2 10.5%

       \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}^{2}}} \]

12. Applied egg-rr 9.9%

    \[\leadsto 1 + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), e^{x}, -1\right)\right)}^{2}}} \]
13. Step-by-step derivation

    1. unpow2 9.9%

       \[\leadsto 1 + \sqrt{\color{blue}{\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), e^{x}, -1\right) \cdot \mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), e^{x}, -1\right)}} \]

    2. rem-sqrt-square 9.9%

       \[\leadsto 1 + \color{blue}{\left|\mathsf{fma}\left(\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), e^{x}, -1\right)\right|} \]

    3. fma-def 9.9%

       \[\leadsto 1 + \left|\color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot e^{x} + -1}\right| \]

    4. *-commutative 9.9%

       \[\leadsto 1 + \left|\color{blue}{e^{x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)} + -1\right| \]

    5. fma-def 9.9%

       \[\leadsto 1 + \left|\color{blue}{\mathsf{fma}\left(e^{x}, \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), -1\right)}\right| \]

14. Simplified 9.9%

    \[\leadsto 1 + \color{blue}{\left|\mathsf{fma}\left(e^{x}, \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right), -1\right)\right|} \]

15. Taylor expanded in x around 0 9.9%

    \[\leadsto 1 + \left|\color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, -0.25 \cdot x, 1\right)\right)\right) - 1}\right| \]

16. Final simplification 9.9%

    \[\leadsto 1 + \left|\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) + -1\right| \]

Alternative 2: 6.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (+ (/ (fmod (exp x) (+ 1.0 (* -0.25 (* x x)))) (exp x)) -1.0)))

C

double code(double x) {
	return 1.0 + ((fmod(exp(x), (1.0 + (-0.25 * (x * x)))) / exp(x)) + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((mod(exp(x), (1.0d0 + ((-0.25d0) * (x * x)))) / exp(x)) + (-1.0d0))
end function

Python

def code(x):
	return 1.0 + ((math.fmod(math.exp(x), (1.0 + (-0.25 * (x * x)))) / math.exp(x)) + -1.0)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / exp(x)) + -1.0))
end

Wolfram

code[x_] := N[(1.0 + N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

   1. expm1-log1p-u 6.3%

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

   2. expm1-udef 6.3%

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

   3. log1p-udef 6.3%

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

   4. add-exp-log 6.3%

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

5. Applied egg-rr 6.3%

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

   1. associate--l+ 6.3%

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

7. Simplified 6.3%

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

8. Taylor expanded in x around 0 5.8%

   \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} - 1\right) \]
9. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} - 1\right) \]

   2. unpow2 5.8%

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

10. Simplified 5.8%

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

11. Final simplification 5.8%

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

Alternative 3: 6.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 1.0 (* -0.25 (* x x)))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (1.0 + (-0.25 * (x * x)))) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), (1.0d0 + ((-0.25d0) * (x * x)))) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), (1.0 + (-0.25 * (x * x)))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in x around 0 5.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
5. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} - 1\right) \]

   2. unpow2 5.8%

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

6. Simplified 5.8%

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

7. Final simplification 5.8%

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

Alternative 4: 6.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[Exp[x], $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 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

   1. expm1-log1p-u 6.3%

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

   2. expm1-udef 6.3%

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

   3. log1p-udef 6.3%

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

   4. add-exp-log 6.3%

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

5. Applied egg-rr 6.3%

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

   1. associate--l+ 6.3%

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

7. Simplified 6.3%

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

   1. associate-+r- 6.3%

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

   2. add-exp-log 6.3%

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

   3. log1p-udef 6.3%

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

   4. expm1-udef 6.3%

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

   5. expm1-log1p-u 6.3%

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

   6. clear-num 6.3%

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

9. Applied egg-rr 6.3%

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

10. Taylor expanded in x around 0 5.6%

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

11. Final simplification 5.6%

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

Alternative 5: 6.5% 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 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in x around 0 5.6%

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

5. Final simplification 5.6%

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

Alternative 6: 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 6.3%

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

   1. exp-neg 6.3%

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

   2. associate-*r/ 6.3%

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

   3. *-rgt-identity 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in x around 0 5.6%

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

5. Taylor expanded in x around 0 4.8%

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

6. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))