expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 6.8%

Time: 18.8s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. x = -5.277175859918918e+240

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: 6.8% 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: 6.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow E (- (log (fmod (exp x) (sqrt (cos x)))) x)))

C

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

Python

def code(x):
	return math.pow(math.e, (math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x)))) - x))

Julia

function code(x)
	return exp(1) ^ Float64(log(rem(exp(x), sqrt(cos(x)))) - x)
end

Wolfram

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

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

   1. add-exp-log 5.5%

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

   2. div-exp 5.5%

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

5. Applied egg-rr 5.5%

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

   1. *-un-lft-identity 5.5%

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

   2. exp-prod 5.5%

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

   3. exp-1-e 5.5%

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

7. Applied egg-rr 5.5%

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

8. Final simplification 5.5%

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

Alternative 2: 6.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

   1. add-exp-log 5.5%

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

   2. div-exp 5.5%

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

5. Applied egg-rr 5.5%

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

6. Final simplification 5.5%

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

Alternative 3: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * Float64(-1.0 / Float64(-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[(-1.0 / (-N[Exp[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

   1. frac-2neg 5.5%

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

   2. div-inv 5.5%

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

5. Applied egg-rr 5.5%

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

6. Final simplification 5.5%

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

Alternative 4: 6.8% 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 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Final simplification 5.5%

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

Alternative 5: 6.6% 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 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Taylor expanded in x around 0 5.3%

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

   1. *-commutative 5.3%

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

   2. unpow2 5.3%

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

6. Simplified 5.3%

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

7. Final simplification 5.3%

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

Alternative 6: 6.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) * Float64(Float64(-(-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[((--1.0) / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

   1. frac-2neg 5.5%

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

   2. div-inv 5.5%

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

5. Applied egg-rr 5.5%

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

6. Taylor expanded in x around 0 5.1%

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

7. Taylor expanded in x around inf 5.1%

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

8. Final simplification 5.1%

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

Alternative 7: 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 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Taylor expanded in x around 0 5.1%

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

5. Final simplification 5.1%

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

Alternative 8: 6.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(-\left(\left(e^{x}\right) \bmod 1\right)\right) \cdot \left(x + \left(-1 + \left(x \cdot x\right) \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (fmod (exp x) 1.0)) (+ x (+ -1.0 (* (* x x) -0.5)))))

C

double code(double x) {
	return -fmod(exp(x), 1.0) * (x + (-1.0 + ((x * x) * -0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -mod(exp(x), 1.0d0) * (x + ((-1.0d0) + ((x * x) * (-0.5d0))))
end function

Python

def code(x):
	return -math.fmod(math.exp(x), 1.0) * (x + (-1.0 + ((x * x) * -0.5)))

Julia

function code(x)
	return Float64(Float64(-rem(exp(x), 1.0)) * Float64(x + Float64(-1.0 + Float64(Float64(x * x) * -0.5))))
end

Wolfram

code[x_] := N[((-N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]) * N[(x + N[(-1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

   1. frac-2neg 5.5%

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

   2. div-inv 5.5%

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

5. Applied egg-rr 5.5%

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

6. Taylor expanded in x around 0 5.1%

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

7. Taylor expanded in x around 0 5.0%

   \[\leadsto \left(-\left(\left(e^{x}\right) \bmod 1\right)\right) \cdot \color{blue}{\left(\left(x + -0.5 \cdot {x}^{2}\right) - 1\right)} \]
8. Step-by-step derivation

   1. associate--l+ 5.0%

      \[\leadsto \left(-\left(\left(e^{x}\right) \bmod 1\right)\right) \cdot \color{blue}{\left(x + \left(-0.5 \cdot {x}^{2} - 1\right)\right)} \]

   2. *-commutative 5.0%

      \[\leadsto \left(-\left(\left(e^{x}\right) \bmod 1\right)\right) \cdot \left(x + \left(\color{blue}{{x}^{2} \cdot -0.5} - 1\right)\right) \]

   3. unpow2 5.0%

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

9. Simplified 5.0%

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

10. Final simplification 5.0%

    \[\leadsto \left(-\left(\left(e^{x}\right) \bmod 1\right)\right) \cdot \left(x + \left(-1 + \left(x \cdot x\right) \cdot -0.5\right)\right) \]

Alternative 9: 5.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Taylor expanded in x around 0 5.1%

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

   1. div-inv 5.1%

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

   2. rec-exp 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around 0 4.9%

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

   1. +-commutative 4.9%

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

   2. *-lft-identity 4.9%

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

   3. associate-*r* 4.9%

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

   4. neg-mul-1 4.9%

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

   5. distribute-rgt-out 4.9%

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

   6. unsub-neg 4.9%

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

9. Simplified 4.9%

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

    1. flip-- 4.9%

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

    2. associate-*r/ 4.9%

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

    3. metadata-eval 4.9%

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

    4. sub-neg 4.9%

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

    5. add-sqr-sqrt 2.6%

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

    6. sqrt-prod 4.9%

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

    7. sqr-neg 4.9%

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

    8. sqrt-unprod 2.3%

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

    9. add-sqr-sqrt 4.9%

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

    10. distribute-lft-neg-out 4.9%

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

    11. sqr-neg 4.9%

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

    12. +-commutative 4.9%

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

11. Applied egg-rr 4.9%

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

12. Final simplification 4.9%

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

Alternative 10: 5.8% 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 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Taylor expanded in x around 0 5.1%

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

   1. div-inv 5.1%

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

   2. rec-exp 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around 0 4.9%

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

   1. +-commutative 4.9%

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

   2. *-lft-identity 4.9%

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

   3. associate-*r* 4.9%

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

   4. neg-mul-1 4.9%

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

   5. distribute-rgt-out 4.9%

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

   6. unsub-neg 4.9%

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

9. Simplified 4.9%

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

10. Final simplification 4.9%

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

Alternative 11: 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 5.5%

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

   1. exp-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. *-rgt-identity 5.5%

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

3. Simplified 5.5%

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

4. Taylor expanded in x around 0 5.1%

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

5. Taylor expanded in x around 0 4.7%

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

6. Final simplification 4.7%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))