expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 45.9%

Time: 8.9s

Alternatives: 7

Speedup: 1.9×

• could not determine a ground truth

  1. x = -5.607202494201913e+123

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 45.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.1)
   (fmod (exp x) (* (* (- (/ 1.0 (* x x)) 0.25) x) x))
   (/ (fmod (- x -1.0) (fma -0.25 (* x x) 1.0)) (exp x))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.1) {
		tmp = fmod(exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x));
	} else {
		tmp = fmod((x - -1.0), fma(-0.25, (x * x), 1.0)) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.1)
		tmp = rem(exp(x), Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x));
	else
		tmp = Float64(rem(Float64(x - -1.0), fma(-0.25, Float64(x * x), 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 0.1], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $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))) < 0.10000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\frac{-1}{4} \cdot {x}^{2}}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{1}{{x}^{2}} - \color{blue}{\frac{1}{4}}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.5%

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

   if 0.10000000000000001 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 19.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

      3. mul0-lft N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      6. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      8. distribute-rgt-out N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      9. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

      10. *-commutative N/A

          \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

      11. +-commutative N/A

          \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(x + \color{blue}{-1 \cdot -1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       3. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       4. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{1} \cdot -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower--.f64 96.1

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

   11. Applied rewrites 96.1%

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

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{e^{-x}} \]

       3. lift-neg.f64 N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

       4. exp-neg N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

       5. lift-exp.f64 N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \frac{1}{\color{blue}{e^{x}}} \]

       6. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot 1}{e^{x}}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot 1}{e^{x}}} \]

       8. lower-*.f64 96.2

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

   13. Applied rewrites 96.2%

       \[\leadsto \color{blue}{\frac{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot 1}{e^{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0.1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(\left(\frac{1}{x \cdot x} - 0.25\right) \cdot x\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.1)
     (fmod (exp x) (* (* (- (/ 1.0 (* x x)) 0.25) x) x))
     (* (fmod (- x -1.0) (fma (* x x) -0.25 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.1) {
		tmp = fmod(exp(x), ((((1.0 / (x * x)) - 0.25) * x) * x));
	} else {
		tmp = fmod((x - -1.0), fma((x * x), -0.25, 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.1)
		tmp = rem(exp(x), Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) - 0.25) * x) * x));
	else
		tmp = Float64(rem(Float64(x - -1.0), fma(Float64(x * x), -0.25, 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.1], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, 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))) < 0.10000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\frac{-1}{4} \cdot {x}^{2}}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{1}{{x}^{2}} - \color{blue}{\frac{1}{4}}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.5%

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

   if 0.10000000000000001 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 19.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

      3. mul0-lft N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      6. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      8. distribute-rgt-out N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      9. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

      10. *-commutative N/A

          \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

      11. +-commutative N/A

          \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(x + \color{blue}{-1 \cdot -1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       3. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       4. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{1} \cdot -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower--.f64 96.1

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

   11. Applied rewrites 96.1%

       \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 25.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(x \cdot x\right) \cdot -0.25\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-163)
   (* (fmod (- x -1.0) (fma (* x x) -0.25 1.0)) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod 1.0 (* (* x x) -0.25)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-163) {
		tmp = fmod((x - -1.0), fma((x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-163)
		tmp = Float64(rem(Float64(x - -1.0), fma(Float64(x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e-163], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999969e-163

   1. Initial program 11.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

      3. mul0-lft N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      6. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      8. distribute-rgt-out N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      9. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

      10. *-commutative N/A

          \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

      11. +-commutative N/A

          \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(x + \color{blue}{-1 \cdot -1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       3. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       4. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{1} \cdot -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

       6. lower--.f64 10.7

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

   11. Applied rewrites 10.7%

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

   12. Taylor expanded in x around 0

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot x - 1\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot x - 1\right) \cdot x} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, 1\right)} \]

       4. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x - \color{blue}{1 \cdot 1}, x, 1\right) \]

       5. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, x, 1\right) \]

       6. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x + \color{blue}{-1} \cdot 1, x, 1\right) \]

       7. metadata-eval N/A

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot x + \color{blue}{-1}, x, 1\right) \]

       8. lower-fma.f64 10.7

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, -1\right)}, x, 1\right) \]

   14. Applied rewrites 10.7%

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)} \]

   if 3.99999999999999969e-163 < x

   1. Initial program 3.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

      2. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

      3. mul0-lft N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

      6. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      8. distribute-rgt-out N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

      9. +-commutative N/A

         \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

      10. *-commutative N/A

          \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

      11. +-commutative N/A

          \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot e^{-x} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{-0.25}\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 25.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 23.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

   3. mul0-lft N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   4. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   5. distribute-lft1-in N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   6. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   7. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   8. distribute-rgt-out N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   9. +-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

   10. *-commutative N/A

       \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

   11. +-commutative N/A

       \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

8. Applied rewrites 23.1%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    2. metadata-eval N/A

       \[\leadsto \left(\left(x + \color{blue}{-1 \cdot -1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    3. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    4. metadata-eval N/A

       \[\leadsto \left(\left(x - \color{blue}{1} \cdot -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    5. metadata-eval N/A

       \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    6. lower--.f64 27.1

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

11. Applied rewrites 27.1%

    \[\leadsto \left(\color{blue}{\left(x - -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]
12. Add Preprocessing

Alternative 5: 6.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 23.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

   3. mul0-lft N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   4. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   5. distribute-lft1-in N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   6. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   7. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   8. distribute-rgt-out N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   9. +-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

   10. *-commutative N/A

       \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

   11. +-commutative N/A

       \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

8. Applied rewrites 23.1%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    2. metadata-eval N/A

       \[\leadsto \left(\left(x + \color{blue}{-1 \cdot -1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    3. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    4. metadata-eval N/A

       \[\leadsto \left(\left(x - \color{blue}{1} \cdot -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    5. metadata-eval N/A

       \[\leadsto \left(\left(x - \color{blue}{-1}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

    6. lower--.f64 27.1

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

11. Applied rewrites 27.1%

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

12. Taylor expanded in x around 0

    \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
13. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)} \]

    2. metadata-eval N/A

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{1} \cdot x\right) \]

    3. *-lft-identity N/A

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

    4. lower--.f64 7.9

       \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

14. Applied rewrites 7.9%

    \[\leadsto \left(\left(x - -1\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]
15. Add Preprocessing

Alternative 6: 4.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 23.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

   3. mul0-lft N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   4. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   5. distribute-lft1-in N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   6. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   7. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   8. distribute-rgt-out N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   9. +-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

   10. *-commutative N/A

       \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

   11. +-commutative N/A

       \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

8. Applied rewrites 23.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
10. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)} \]

    2. metadata-eval N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{1} \cdot x\right) \]

    3. *-lft-identity N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

    4. lower--.f64 4.1

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

11. Applied rewrites 4.1%

    \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]
12. Add Preprocessing

Alternative 7: 3.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 23.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(1 \bmod \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2} + 1\right)}\right) \cdot e^{-x} \]

   2. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(0 + 1\right)}\right)\right) \cdot e^{-x} \]

   3. mul0-lft N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{0 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   4. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 + 1\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   5. distribute-lft1-in N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + 1\right)\right)\right) \cdot e^{-x} \]

   6. *-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{2}\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   7. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   8. distribute-rgt-out N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)\right)\right) \cdot e^{-x} \]

   9. +-commutative N/A

      \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot {x}^{2} + \color{blue}{\left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)}\right)\right) \cdot e^{-x} \]

   10. *-commutative N/A

       \[\leadsto \left(1 \bmod \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(1 + \left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \cdot e^{-x} \]

   11. +-commutative N/A

       \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \frac{-1}{4} + \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \frac{1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + 1\right)}\right)\right) \cdot e^{-x} \]

8. Applied rewrites 23.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
10. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)} \]

    2. metadata-eval N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{1} \cdot x\right) \]

    3. *-lft-identity N/A

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

    4. lower--.f64 4.1

       \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

11. Applied rewrites 4.1%

    \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

12. Taylor expanded in x around inf

    \[\leadsto \left(1 \bmod \left(\frac{-1}{4} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot \left(1 - x\right) \]
13. Step-by-step derivation

14. Applied rewrites 3.2%

    \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{-0.25}\right)\right) \cdot \left(1 - x\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))