expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 7.0%

Time: 20.5s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. x = -5.578438092639143e+108

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 7.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[Log[N[Exp[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]], $MachinePrecision]

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

   1. add-log-exp 8.7%

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

5. Applied egg-rr 8.7%

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

6. Final simplification 8.7%

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

Alternative 2: 7.0% 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

   1. add-exp-log 8.6%

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

   2. div-exp 8.6%

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

5. Applied egg-rr 8.6%

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

6. Final simplification 8.6%

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

Alternative 3: 7.0% 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

   1. expm1-log1p-u 8.6%

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

   2. expm1-udef 8.6%

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

5. Applied egg-rr 8.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1} \]
6. Step-by-step derivation

   1. sub-neg 8.6%

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

   2. log1p-udef 8.6%

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

   3. add-exp-log 8.6%

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

   4. +-commutative 8.6%

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

   5. metadata-eval 8.6%

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

7. Applied egg-rr 8.6%

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

8. Final simplification 8.6%

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

Alternative 4: 7.0% 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

   1. add-log-exp 8.7%

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

5. Applied egg-rr 8.7%

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

   1. add-log-exp 8.6%

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

   2. clear-num 8.6%

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

7. Applied egg-rr 8.6%

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

8. Final simplification 8.6%

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

Alternative 5: 7.0% 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Final simplification 8.6%

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

Alternative 6: 6.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(1 + \left(-1 + {x}^{2} \cdot -0.25\right)\right)\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 1.0 (+ 1.0 (+ -1.0 (* (pow x 2.0) -0.25))))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (1.0 + (1.0 + (-1.0 + (pow(x, 2.0) * -0.25))))) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), (1.0d0 + (1.0d0 + ((-1.0d0) + ((x ** 2.0d0) * (-0.25d0)))))) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), (1.0 + (1.0 + (-1.0 + (math.pow(x, 2.0) * -0.25))))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), Float64(1.0 + Float64(1.0 + Float64(-1.0 + Float64((x ^ 2.0) * -0.25))))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(1.0 + N[(-1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 8.1%

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

   1. *-commutative 7.4%

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

6. Simplified 8.1%

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

   1. expm1-log1p-u 7.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2} \cdot -0.25\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

   2. log1p-def 7.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {x}^{2} \cdot -0.25\right)}\right)\right)\right) \cdot \left(1 - x\right) \]

   3. expm1-udef 7.3%

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

   4. add-exp-log 7.4%

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

   5. associate--l+ 7.4%

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

8. Applied egg-rr 8.1%

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

9. Final simplification 8.1%

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

Alternative 7: 6.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (exp x) (fmod (exp x) (+ 1.0 (* (pow x 2.0) -0.25))))))

C

double code(double x) {
	return 1.0 / (exp(x) / fmod(exp(x), (1.0 + (pow(x, 2.0) * -0.25))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (exp(x) / mod(exp(x), (1.0d0 + ((x ** 2.0d0) * (-0.25d0)))))
end function

Python

def code(x):
	return 1.0 / (math.exp(x) / math.fmod(math.exp(x), (1.0 + (math.pow(x, 2.0) * -0.25))))

Julia

function code(x)
	return Float64(1.0 / Float64(exp(x) / rem(exp(x), Float64(1.0 + Float64((x ^ 2.0) * -0.25)))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

   1. add-log-exp 8.7%

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

5. Applied egg-rr 8.7%

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

   1. add-log-exp 8.6%

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

   2. clear-num 8.6%

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

7. Applied egg-rr 8.6%

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

8. Taylor expanded in x around 0 8.1%

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

   1. *-commutative 7.4%

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

10. Simplified 8.1%

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

11. Final simplification 8.1%

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

Alternative 8: 6.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 1.0 (* (pow x 2.0) -0.25))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (1.0 + (pow(x, 2.0) * -0.25))) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), (1.0d0 + ((x ** 2.0d0) * (-0.25d0)))) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), (1.0 + (math.pow(x, 2.0) * -0.25))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), Float64(1.0 + Float64((x ^ 2.0) * -0.25))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 8.1%

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

   1. *-commutative 7.4%

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

6. Simplified 8.1%

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

7. Final simplification 8.1%

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

Alternative 9: 6.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 7.5%

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

   1. add-exp-log 7.5%

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

   2. div-exp 7.5%

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

6. Applied egg-rr 7.5%

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

7. Final simplification 7.5%

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

Alternative 10: 6.6% 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 7.5%

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

5. Final simplification 7.5%

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

Alternative 11: 6.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 7.4%

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

   1. +-commutative 7.4%

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

   2. *-lft-identity 7.4%

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

   3. associate-*r* 7.4%

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

   4. neg-mul-1 7.4%

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

   5. distribute-rgt-out 7.4%

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

   6. neg-sub0 7.4%

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

   7. associate-+r- 7.4%

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

   8. metadata-eval 7.4%

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

6. Simplified 7.4%

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

7. Taylor expanded in x around 0 7.4%

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

   1. *-commutative 7.4%

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

9. Simplified 7.4%

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

    1. expm1-log1p-u 7.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2} \cdot -0.25\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

    2. log1p-def 7.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {x}^{2} \cdot -0.25\right)}\right)\right)\right) \cdot \left(1 - x\right) \]

    3. expm1-udef 7.3%

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

    4. add-exp-log 7.4%

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

    5. associate--l+ 7.4%

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

11. Applied egg-rr 7.4%

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

    1. associate-+r- 7.4%

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

    2. add-exp-log 7.3%

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

    3. expm1-def 7.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\mathsf{expm1}\left(\log \left(1 + {x}^{2} \cdot -0.25\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

    4. log1p-def 7.3%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({x}^{2} \cdot -0.25\right)}\right)\right)\right) \cdot \left(1 - x\right) \]

    5. expm1-log1p-u 7.4%

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

    6. *-commutative 7.4%

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

    7. unpow2 7.4%

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

    8. associate-*r* 7.4%

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

13. Applied egg-rr 7.4%

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

14. Final simplification 7.4%

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

Alternative 12: 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 7.5%

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

5. Taylor expanded in x around 0 7.2%

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

   1. +-commutative 7.2%

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

   2. *-lft-identity 7.2%

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

   3. associate-*r* 7.2%

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

   4. neg-mul-1 7.2%

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

   5. distribute-rgt-out 7.1%

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

   6. sub-neg 7.1%

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

7. Simplified 7.1%

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

8. Final simplification 7.1%

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

Alternative 13: 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 8.6%

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

   1. /-rgt-identity 8.6%

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

   2. associate-/r/ 8.6%

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

   3. exp-neg 8.6%

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

   4. remove-double-neg 8.6%

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

3. Simplified 8.6%

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

4. Taylor expanded in x around 0 7.5%

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

5. Taylor expanded in x around 0 6.5%

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

6. Final simplification 6.5%

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

Reproduce

herbie shell --seed 2023326 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))