expfmod (used to be hard to sample)

Percentage Accurate: 8.5% → 8.5%

Time: 57.3s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = -3.964449964662712e-204

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: 8.5% 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: 8.5% 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 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

   1. expm1-log1p-u 7.8%

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

   2. expm1-udef 7.8%

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

   3. log1p-udef 7.8%

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

   4. add-exp-log 7.8%

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

5. Applied egg-rr 7.8%

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

6. Final simplification 7.8%

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

Alternative 2: 8.5% 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 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

   1. expm1-log1p-u 7.8%

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

   2. expm1-udef 7.8%

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

   3. log1p-udef 7.8%

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

   4. add-exp-log 7.8%

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

5. Applied egg-rr 7.8%

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

   1. add-exp-log 7.8%

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

   2. log1p-udef 7.8%

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

   3. expm1-udef 7.8%

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

   4. expm1-log1p-u 7.8%

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

   5. clear-num 7.8%

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

7. Applied egg-rr 7.8%

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

8. Final simplification 7.8%

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

Alternative 3: 8.5% 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 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

4. Final simplification 7.8%

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

Alternative 4: 7.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Derivation

1. Initial program 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in x around 0 7.0%

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

   1. mul-1-neg 7.0%

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

   2. unsub-neg 7.0%

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

   3. *-rgt-identity 7.0%

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

   4. distribute-lft-out-- 7.0%

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

6. Simplified 7.0%

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

   1. flip-- 7.0%

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

   2. associate-*r/ 7.0%

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

   3. metadata-eval 7.0%

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

8. Applied egg-rr 7.0%

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

   1. associate-/l* 7.0%

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

   2. +-commutative 7.0%

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

10. Simplified 7.0%

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

11. Final simplification 7.0%

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

Alternative 5: 7.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Derivation

1. Initial program 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in x around 0 7.0%

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

   1. mul-1-neg 7.0%

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

   2. unsub-neg 7.0%

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

   3. *-rgt-identity 7.0%

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

   4. distribute-lft-out-- 7.0%

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

6. Simplified 7.0%

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

7. Final simplification 7.0%

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

Alternative 6: 7.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in x around 0 6.8%

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

5. Final simplification 6.8%

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

Alternative 7: 5.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. exp-neg 7.8%

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

   2. associate-*r/ 7.8%

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

   3. *-rgt-identity 7.8%

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

3. Simplified 7.8%

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

   1. add-exp-log 7.8%

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

   2. div-exp 7.8%

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

5. Applied egg-rr 7.8%

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

6. Taylor expanded in x around inf 5.2%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
7. Step-by-step derivation

   1. neg-mul-1 5.2%

      \[\leadsto e^{\color{blue}{-x}} \]

8. Simplified 5.2%

   \[\leadsto e^{\color{blue}{-x}} \]

9. Final simplification 5.2%

   \[\leadsto e^{-x} \]

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))))