expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 7.1%

Time: 19.6s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. x = -4.206812028177403e+51

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: 7.1% 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: 7.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. add-exp-log 7.7%

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

   2. div-exp 7.7%

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

5. Applied egg-rr 7.7%

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

   1. add-cbrt-cube 7.8%

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

   2. pow1/3 7.7%

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

   3. add-sqr-sqrt 7.7%

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

   4. pow1 7.7%

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

   5. pow1/2 7.7%

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

   6. pow-prod-up 7.7%

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

   7. metadata-eval 7.7%

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

7. Applied egg-rr 7.7%

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

   1. unpow1/3 7.8%

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

9. Simplified 7.8%

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

10. Final simplification 7.8%

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

Alternative 2: 7.1% 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 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. add-cbrt-cube 7.8%

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

   2. pow1/3 7.7%

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

   3. add-sqr-sqrt 7.7%

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

   4. pow1 7.7%

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

   5. pow1/2 7.7%

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

   6. pow-prod-up 7.7%

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

   7. metadata-eval 7.7%

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

5. Applied egg-rr 7.7%

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

   1. unpow1/3 7.8%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 3: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. expm1-log1p-u 7.7%

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

   2. expm1-udef 7.7%

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

   3. log1p-udef 7.7%

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

   4. add-exp-log 7.7%

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

5. Applied egg-rr 7.7%

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

   1. associate--l+ 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 4: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. expm1-log1p-u 7.7%

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

   2. expm1-udef 7.7%

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

   3. log1p-udef 7.7%

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

   4. add-exp-log 7.7%

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

5. Applied egg-rr 7.7%

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

6. Final simplification 7.7%

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

Alternative 5: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. expm1-log1p-u 7.7%

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

   2. expm1-udef 7.7%

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

   3. log1p-udef 7.7%

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

   4. add-exp-log 7.7%

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

5. Applied egg-rr 7.7%

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

   1. add-exp-log 7.7%

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

   2. log1p-udef 7.7%

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

   3. expm1-udef 7.7%

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

   4. expm1-log1p-u 7.7%

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

   5. clear-num 7.7%

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

7. Applied egg-rr 7.7%

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

8. Final simplification 7.7%

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

Alternative 6: 7.1% 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 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

4. Final simplification 7.7%

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

Alternative 7: 6.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

   1. add-exp-log 7.7%

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

   2. div-exp 7.7%

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

5. Applied egg-rr 7.7%

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

6. Taylor expanded in x around 0 7.2%

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

   1. *-commutative 7.2%

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

   2. unpow2 7.2%

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

8. Simplified 7.2%

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

9. Final simplification 7.2%

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

Alternative 8: 6.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

4. Taylor expanded in x around 0 7.2%

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

   1. *-commutative 7.2%

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

   2. unpow2 7.2%

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

6. Simplified 7.2%

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

7. Final simplification 7.2%

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

Alternative 9: 6.7% 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 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

4. Taylor expanded in x around 0 6.7%

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

5. Final simplification 6.7%

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

Alternative 10: 5.5% 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 7.7%

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

   1. exp-neg 7.7%

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

   2. associate-*r/ 7.7%

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

   3. *-rgt-identity 7.7%

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

3. Simplified 7.7%

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

4. Taylor expanded in x around 0 6.7%

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

5. Taylor expanded in x around 0 5.8%

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

6. Final simplification 5.8%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))