expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 60.8%

Time: 11.8s

Alternatives: 10

Speedup: 3.9×

• could not determine a ground truth

  1. x = -1.4936278584526992e+301

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 60.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \sqrt{\cos x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod t\_1\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod t\_1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (sqrt (cos x))))
   (if (<= (* t_0 (fmod (exp x) t_1)) 0.02)
     (* (fmod (* (fma 0.5 x 1.0) x) t_1) t_0)
     (* (fmod (- x -1.0) t_1) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = sqrt(cos(x));
	double tmp;
	if ((t_0 * fmod(exp(x), t_1)) <= 0.02) {
		tmp = fmod((fma(0.5, x, 1.0) * x), t_1) * t_0;
	} else {
		tmp = fmod((x - -1.0), t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = sqrt(cos(x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), t_1)) <= 0.02)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), t_1) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $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))) < 0.0200000000000000004

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.8

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

   8. Applied rewrites 5.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

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

   12. Taylor expanded in x around inf

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

       1. lower-sqrt.f64 N/A

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

       2. lower-cos.f64 50.1

          \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot e^{-x} \]

   14. Applied rewrites 50.1%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot e^{-x} \]

   if 0.0200000000000000004 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \sqrt{\cos x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod t\_1\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5 \cdot x, x, x\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (sqrt (cos x))))
   (if (<= (* t_0 (fmod (exp x) t_1)) 0.02)
     (* (fmod (fma (* 0.5 x) x x) 1.0) t_0)
     (* (fmod (- x -1.0) t_1) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = sqrt(cos(x));
	double tmp;
	if ((t_0 * fmod(exp(x), t_1)) <= 0.02) {
		tmp = fmod(fma((0.5 * x), x, x), 1.0) * t_0;
	} else {
		tmp = fmod((x - -1.0), t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = sqrt(cos(x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), t_1)) <= 0.02)
		tmp = Float64(rem(fma(Float64(0.5 * x), x, x), 1.0) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[With[{TMP1 = N[(N[(0.5 * x), $MachinePrecision] * x + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $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))) < 0.0200000000000000004

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.8

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

   8. Applied rewrites 5.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.1%

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

   if 0.0200000000000000004 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

      \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5 \cdot x, x, x\right)\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5 \cdot x, x, x\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 0.02)
     (* (fmod (fma (* 0.5 x) x x) 1.0) t_0)
     (* (fmod (- x -1.0) 1.0) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 0.02) {
		tmp = fmod(fma((0.5 * x), x, x), 1.0) * t_0;
	} else {
		tmp = fmod((x - -1.0), 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 0.02)
		tmp = Float64(rem(fma(Float64(0.5 * x), x, x), 1.0) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(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]), $MachinePrecision], 0.02], N[(N[With[{TMP1 = N[(N[(0.5 * x), $MachinePrecision] * x + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $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))) < 0.0200000000000000004

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.8

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

   8. Applied rewrites 5.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.1%

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

   if 0.0200000000000000004 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5 \cdot x, x, x\right)\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 0.02)
     (* (fmod (* (fma 0.5 x 1.0) x) 1.0) t_0)
     (* (fmod (- x -1.0) 1.0) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 0.02) {
		tmp = fmod((fma(0.5, x, 1.0) * x), 1.0) * t_0;
	} else {
		tmp = fmod((x - -1.0), 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 0.02)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), 1.0) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(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]), $MachinePrecision], 0.02], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $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))) < 0.0200000000000000004

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.8

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

   8. Applied rewrites 5.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

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

   if 0.0200000000000000004 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.02:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 25.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 2.0)
   (*
    (fmod
     (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)
     (sqrt (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0)))
    (fma x (fma (fma -0.16666666666666666 x 0.5) x -1.0) 1.0))
   (* 1.0 (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0), sqrt(fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0))) * fma(x, fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), 1.0);
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0), sqrt(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0))) * fma(x, fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), 1.0));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 2.0], N[(N[With[{TMP1 = N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 9.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]

   5. Applied rewrites 8.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.3%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), 1\right) \cdot \left(\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 8.2%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \]

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.0%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 25.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. lower--.f64 24.5

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.4%

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

Alternative 7: 25.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (*
    (fmod
     (fma (fma 0.5 x 1.0) x 1.0)
     (sqrt (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0)))
    (fma x (fma (fma -0.16666666666666666 x 0.5) x -1.0) 1.0))
   (* 1.0 (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), sqrt(fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0))) * fma(x, fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), 1.0);
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), sqrt(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0))) * fma(x, fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), 1.0));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]

   5. Applied rewrites 8.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 8.4%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right), x, -1\right), 1\right) \cdot \left(\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 8.2%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \]

   if 0.5 < x

   1. Initial program 2.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.0%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50.0)
   (* (fma (fma 0.5 x -1.0) x 1.0) (fmod (fma (fma 0.5 x 1.0) x 1.0) 1.0))
   (* 1.0 (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 50.0) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * fmod(fma(fma(0.5, x, 1.0), x, 1.0), 1.0);
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 50.0)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * rem(fma(fma(0.5, x, 1.0), x, 1.0), 1.0));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 50.0], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 9.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 8.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 8.4

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 7.9

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

   10. Applied rewrites 7.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, -1\right), x, 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 8.0

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   13. Applied rewrites 8.0%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \]

   if 50 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.1%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 23.0

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

11. Applied rewrites 23.0%

    \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot 1 \]
12. Add Preprocessing

Alternative 10: 22.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 21.4%

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

12. Final simplification 21.4%

    \[\leadsto 1 \cdot \left(1 \bmod 1\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))