expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 26.0%

Time: 15.8s

Alternatives: 4

Speedup: 5.0×

• could not determine a ground truth

  1. x = -5.99919578068032e+67

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: 6.7% 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: 26.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (t_0 * math.exp(-x)) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-x))) <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 7.5%

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

      1. /-rgt-identity 7.5%

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

      2. associate-/r/ 7.5%

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

      3. exp-neg 7.5%

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

      4. remove-double-neg 7.5%

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

   3. Simplified 7.5%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 0.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]

      2. cos-0 100.0%

         \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \]

      3. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 2: 25.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fmod(exp(x), sqrt(cos(x))) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) * (1.0 - x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(rem(exp(x), sqrt(cos(x))) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 7.4%

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

   2. Taylor expanded in x around 0 6.8%

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

      1. associate-*r* 6.8%

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

      2. neg-mul-1 6.8%

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

      3. distribute-lft1-in 6.8%

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

   4. Simplified 6.8%

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

   5. Taylor expanded in x around 0 6.8%

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

      1. +-commutative 6.8%

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

      2. mul-1-neg 6.8%

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

      3. unsub-neg 6.8%

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

      4. *-lft-identity 6.8%

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

      5. distribute-rgt-out-- 6.8%

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

   7. Simplified 6.8%

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

   if 1 < x

   1. Initial program 0.7%

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

   2. Taylor expanded in x around 0 0.2%

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

   4. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 96.4%

         \[\leadsto \color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]

      2. cos-0 96.4%

         \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \]

      3. metadata-eval 96.4%

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

   6. Simplified 96.4%

      \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 3: 24.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50.0) (fmod (exp x) (sqrt (cos x))) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 50.0) {
		tmp = fmod(exp(x), sqrt(cos(x)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 50.0d0) then
        tmp = mod(exp(x), sqrt(cos(x)))
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 50.0:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 50.0)
		tmp = rem(exp(x), sqrt(cos(x)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 50.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 6.4%

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

   if 50 < x

   1. Initial program 0.4%

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

   2. Taylor expanded in x around 0 0.1%

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

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 98.2%

         \[\leadsto \color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]

      2. cos-0 98.2%

         \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \]

      3. metadata-eval 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]

Alternative 4: 22.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod 1.0 1.0))

C

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

Fortran

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

Python

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

Julia

function code(x)
	return rem(1.0, 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 6.0%

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

2. Taylor expanded in x around 0 5.1%

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

4. Applied egg-rr 22.9%

   \[\leadsto \color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 22.9%

      \[\leadsto \color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)} \]

   2. cos-0 22.9%

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \]

   3. metadata-eval 22.9%

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

6. Simplified 22.9%

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

7. Final simplification 22.9%

   \[\leadsto \left(1 \bmod 1\right) \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))