expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 9.4%

Time: 18.8s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = -1.7978738927178525e+32

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 9.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}\\ \frac{\mathsf{fma}\left(t_0, t_0, -1\right) + -1}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (fmod (exp x) (sqrt (cos x))) 2.0))))
   (/ (+ (fma t_0 t_0 -1.0) -1.0) (exp x))))

C

double code(double x) {
	double t_0 = sqrt((fmod(exp(x), sqrt(cos(x))) + 2.0));
	return (fma(t_0, t_0, -1.0) + -1.0) / exp(x);
}

Julia

function code(x)
	t_0 = sqrt(Float64(rem(exp(x), sqrt(cos(x))) + 2.0))
	return Float64(Float64(fma(t_0, t_0, -1.0) + -1.0) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

   1. expm1-log1p-u 5.7%

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

   2. expm1-udef 5.8%

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

   3. log1p-udef 5.8%

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

   4. rem-exp-log 5.8%

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

5. Applied egg-rr 5.8%

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

   1. expm1-log1p-u 5.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} - 1}{e^{x}} \]

   2. expm1-udef 5.7%

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

   3. log1p-udef 5.7%

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

   4. +-commutative 5.7%

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

   5. add-exp-log 5.7%

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

   6. add-sqr-sqrt 8.5%

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

   7. fma-neg 8.5%

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

7. Applied egg-rr 8.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, -1\right)} - 1}{e^{x}} \]

8. Final simplification 8.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, -1\right) + -1}{e^{x}} \]

Alternative 2: 9.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}\\ \frac{\mathsf{fma}\left(t_0, t_0, -2\right)}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (fmod (exp x) (sqrt (cos x))) 2.0))))
   (/ (fma t_0 t_0 -2.0) (exp x))))

C

double code(double x) {
	double t_0 = sqrt((fmod(exp(x), sqrt(cos(x))) + 2.0));
	return fma(t_0, t_0, -2.0) / exp(x);
}

Julia

function code(x)
	t_0 = sqrt(Float64(rem(exp(x), sqrt(cos(x))) + 2.0))
	return Float64(fma(t_0, t_0, -2.0) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + -2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

   1. expm1-log1p-u 5.7%

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

   2. expm1-udef 5.8%

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

   3. log1p-udef 5.8%

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

   4. rem-exp-log 5.8%

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

5. Applied egg-rr 5.8%

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

   1. expm1-log1p-u 5.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} - 1}{e^{x}} \]

   2. expm1-udef 5.7%

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

   3. log1p-udef 5.7%

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

   4. +-commutative 5.7%

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

   5. add-exp-log 5.7%

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

   6. add-sqr-sqrt 8.5%

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

   7. fma-neg 8.5%

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

7. Applied egg-rr 8.5%

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

   1. fma-udef 8.5%

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

   2. difference-of-sqr--1 8.5%

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

9. Applied egg-rr 8.5%

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

    1. difference-of-sqr--1 8.5%

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

    2. add-sqr-sqrt 5.7%

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

    3. associate--l+ 5.7%

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

    4. metadata-eval 5.7%

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

    5. add-sqr-sqrt 8.5%

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

    6. fma-def 8.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, -2\right)}}{e^{x}} \]

11. Applied egg-rr 8.5%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, -2\right)}}{e^{x}} \]

12. Final simplification 8.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}, -2\right)}{e^{x}} \]

Alternative 3: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 2}\\ \frac{\left(t_0 + 1\right) \cdot \left(t_0 + -1\right) + -1}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (fmod (exp x) (sqrt (cos x))) 2.0))))
   (/ (+ (* (+ t_0 1.0) (+ t_0 -1.0)) -1.0) (exp x))))

C

double code(double x) {
	double t_0 = sqrt((fmod(exp(x), sqrt(cos(x))) + 2.0));
	return (((t_0 + 1.0) * (t_0 + -1.0)) + -1.0) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((mod(exp(x), sqrt(cos(x))) + 2.0d0))
    code = (((t_0 + 1.0d0) * (t_0 + (-1.0d0))) + (-1.0d0)) / exp(x)
end function

Python

def code(x):
	t_0 = math.sqrt((math.fmod(math.exp(x), math.sqrt(math.cos(x))) + 2.0))
	return (((t_0 + 1.0) * (t_0 + -1.0)) + -1.0) / math.exp(x)

Julia

function code(x)
	t_0 = sqrt(Float64(rem(exp(x), sqrt(cos(x))) + 2.0))
	return Float64(Float64(Float64(Float64(t_0 + 1.0) * Float64(t_0 + -1.0)) + -1.0) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

   1. expm1-log1p-u 5.7%

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

   2. expm1-udef 5.8%

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

   3. log1p-udef 5.8%

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

   4. rem-exp-log 5.8%

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

5. Applied egg-rr 5.8%

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

   1. expm1-log1p-u 5.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} - 1}{e^{x}} \]

   2. expm1-udef 5.7%

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

   3. log1p-udef 5.7%

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

   4. +-commutative 5.7%

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

   5. add-exp-log 5.7%

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

   6. add-sqr-sqrt 8.5%

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

   7. fma-neg 8.5%

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

7. Applied egg-rr 8.5%

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

   1. fma-udef 8.5%

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

   2. difference-of-sqr--1 8.5%

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

9. Applied egg-rr 8.5%

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

10. Final simplification 8.5%

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

Alternative 4: 6.9% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

   1. add-log-exp 5.8%

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

5. Applied egg-rr 5.8%

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

6. Final simplification 5.8%

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

Alternative 5: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

   1. expm1-log1p-u 5.7%

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

   2. expm1-udef 5.8%

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

   3. log1p-udef 5.8%

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

   4. rem-exp-log 5.8%

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

5. Applied egg-rr 5.8%

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

6. Final simplification 5.8%

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

Alternative 6: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

   1. exp-neg 5.7%

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

   2. associate-*r/ 5.8%

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

   3. *-rgt-identity 5.8%

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

3. Simplified 5.8%

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

4. Final simplification 5.8%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))