expfmod (used to be hard to sample)

Percentage Accurate: 9.3% → 47.7%

Time: 17.4s

Alternatives: 10

Speedup: 1.5×

• could not determine a ground truth

  1. x = -4.957060683703297e-142

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: 9.3% 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: 47.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\sin \left(\pi \cdot 0.5 - x\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) t_0)
     (if (<= t_1 2.0)
       (* (fmod (exp x) (sqrt (sin (- (* PI 0.5) x)))) t_0)
       (*
        (fmod 1.0 (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0))
        t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), sqrt(sin(((((double) M_PI) * 0.5) - x)))) * t_0;
	} else {
		tmp = fmod(1.0, fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(sin(Float64(Float64(pi * 0.5) - x)))) * t_0);
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 0.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Sin[N[(N[(Pi * 0.5), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unswap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 8.7

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

   6. Applied rewrites 8.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 11.4

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

   9. Applied rewrites 11.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. add-flip 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. metadata-eval N/A

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

      4. lower--.f64 38.9

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

   4. Applied rewrites 38.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lift-neg.f64 N/A

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

      4. sin-+PI/2-rev N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 23.4

          \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\sin \left(\left(-x\right) + \color{blue}{\pi} \cdot 0.5\right)}\right)\right) \cdot e^{-x} \]

   6. Applied rewrites 23.4%

      \[\leadsto \left(\left(x - -1\right) \bmod \left(\sqrt{\color{blue}{\sin \left(\left(-x\right) + \pi \cdot 0.5\right)}}\right)\right) \cdot e^{-x} \]

   7. Taylor expanded in x around -inf

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

      1. sin-sum N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. sin-sum-rev N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. sub-flip N/A

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

      10. sub-flip N/A

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

      11. lift-neg.f64 N/A

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

   9. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\sin \left(\pi \cdot 0.5 - x\right)}\right)\right)} \cdot e^{-x} \]

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

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. sub-flip N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{-x} \]

      6. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      7. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      8. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{-x} \]

      10. lift-*.f64 35.4

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 47.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 0.0)
     (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) t_1)
     (if (<= t_2 2.0)
       (/ t_0 (exp x))
       (*
        (fmod 1.0 (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0))
        t_1)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * t_1;
	} else if (t_2 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0)) * t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_1);
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unswap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 8.7

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

   6. Applied rewrites 8.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 11.4

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

   9. Applied rewrites 11.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 9.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-fmod.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-fmod.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. lift-exp.f64 9.3

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

   3. Applied rewrites 9.3%

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

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

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. sub-flip N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{-x} \]

      6. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      7. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      8. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{-x} \]

      10. lift-*.f64 35.4

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 45.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - -1\right) \bmod t\_0\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) t_0) t_1) 0.0)
     (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) t_1)
     (/ (fmod (- x -1.0) t_0) (exp x)))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), t_0) * t_1) <= 0.0) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * t_1;
	} else {
		tmp = fmod((x - -1.0), t_0) / exp(x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], 0.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = t$95$0}, 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))) < 0.0

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unswap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 8.7

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

   6. Applied rewrites 8.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 11.4

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

   9. Applied rewrites 11.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. add-flip 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. metadata-eval N/A

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

      4. lower--.f64 38.9

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

   4. Applied rewrites 38.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 38.9

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

   6. Applied rewrites 38.9%

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

Alternative 4: 45.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, -0.5\right), 1\right)}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 0.0)
     (* (fmod (exp x) (* (fma 0.5 x 1.0) (* -0.5 x))) t_0)
     (/
      (fmod
       (- x -1.0)
       (sqrt (fma (* x x) (fma (* 0.041666666666666664 x) x -0.5) 1.0)))
      (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 0.0) {
		tmp = fmod(exp(x), (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	} else {
		tmp = fmod((x - -1.0), sqrt(fma((x * x), fma((0.041666666666666664 * x), x, -0.5), 1.0))) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 0.0)
		tmp = Float64(rem(exp(x), Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_0);
	else
		tmp = Float64(rem(Float64(x - -1.0), sqrt(fma(Float64(x * x), fma(Float64(0.041666666666666664 * x), x, -0.5), 1.0))) / exp(x));
	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], 0.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(0.041666666666666664 * x), $MachinePrecision] * x + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $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))) < 0.0

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unswap-sqr N/A

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

      9. metadata-eval N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 8.7

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

   6. Applied rewrites 8.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 11.4

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

   9. Applied rewrites 11.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. add-flip 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. metadata-eval N/A

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

      4. lower--.f64 38.9

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

   4. Applied rewrites 38.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 38.9

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right)}{e^{x}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right)}{e^{x}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}\right) + 1}\right)\right)}{e^{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right) + 1}\right)\right)}{e^{x}} \]

      4. add-flip N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{-1}{2}\right) + 1}\right)\right)}{e^{x}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{-1}{2}}, 1\right)}\right)\right)}{e^{x}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right)} + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right)} + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{24} \cdot x\right) \cdot x + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{24} \cdot x, \color{blue}{x}, \frac{-1}{2}\right), 1\right)}\right)\right)}{e^{x}} \]

      10. lower-*.f64 38.9

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

   9. Applied rewrites 38.9%

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, -0.5\right), 1\right)}}\right)\right)}{e^{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (/ (fmod (exp x) (fma (* x x) -0.25 1.0)) (exp x))
     (*
      (fmod 1.0 (fma (* x x) (fma (* x x) -0.010416666666666666 -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) <= 2.0) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, fma((x * x), fma((x * x), -0.010416666666666666, -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) <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) / exp(x));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $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], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] + 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))) < 2

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 8.8

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

   6. Applied rewrites 8.8%

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

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

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. sub-flip N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{-x} \]

      6. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      7. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      8. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{-x} \]

      10. lift-*.f64 35.4

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (fma (fma 0.5 x -1.0) x 1.0))
   (*
    (fmod 1.0 (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[With[{TMP1 = N[Exp[x], $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] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $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 x < 1.1000000000000001

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 8.1

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

   7. Applied rewrites 8.1%

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

   if 1.1000000000000001 < x

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. sub-flip N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{-x} \]

      6. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      7. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      8. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{-x} \]

      10. lift-*.f64 35.4

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 39.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (* (fmod (exp x) (fma (* x x) -0.25 1.0)) (- 1.0 x))
   (*
    (fmod 1.0 (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0))
    (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(1.0, fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.88], N[(N[With[{TMP1 = N[Exp[x], $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], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $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 x < 0.880000000000000004

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.7

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

   4. Applied rewrites 8.7%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. sub-flip-reverse N/A

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

      3. lower--.f64 7.7

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

   7. Applied rewrites 7.7%

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

   if 0.880000000000000004 < x

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2}} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      5. sub-flip N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{-x} \]

      6. pow2 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{96} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      7. *-commutative N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right), 1\right)\right)\right) \cdot e^{-x} \]

      8. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-1}{96} + \frac{-1}{4}, 1\right)\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{-x} \]

      10. lift-*.f64 35.4

          \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 38.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, -0.5\right), 1\right)}\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fmod
   (- x -1.0)
   (sqrt (fma (* x x) (fma (* 0.041666666666666664 x) x -0.5) 1.0)))
  (exp x)))

C

double code(double x) {
	return fmod((x - -1.0), sqrt(fma((x * x), fma((0.041666666666666664 * x), x, -0.5), 1.0))) / exp(x);
}

Julia

function code(x)
	return Float64(rem(Float64(x - -1.0), sqrt(fma(Float64(x * x), fma(Float64(0.041666666666666664 * x), x, -0.5), 1.0))) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. add-flip 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. metadata-eval N/A

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

   4. lower--.f64 38.9

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

4. Applied rewrites 38.9%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. mult-flip-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-exp.f64 38.9

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

6. Applied rewrites 38.9%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right)}{e^{x}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}}\right)\right)}{e^{x}} \]

   2. pow2 N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}\right) + 1}\right)\right)}{e^{x}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right) + 1}\right)\right)}{e^{x}} \]

   4. add-flip N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{-1}{2}\right) + 1}\right)\right)}{e^{x}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{-1}{2}}, 1\right)}\right)\right)}{e^{x}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right)} + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \left(x \cdot x\right)} + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{24} \cdot x\right) \cdot x + \frac{-1}{2}, 1\right)}\right)\right)}{e^{x}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{24} \cdot x, \color{blue}{x}, \frac{-1}{2}\right), 1\right)}\right)\right)}{e^{x}} \]

   10. lower-*.f64 38.9

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

9. Applied rewrites 38.9%

   \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, -0.5\right), 1\right)}}\right)\right)}{e^{x}} \]
10. Add Preprocessing

Alternative 9: 7.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $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 9.3%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 8.7

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

4. Applied rewrites 8.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. sub-flip-reverse N/A

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

   3. lower--.f64 7.7

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

7. Applied rewrites 7.7%

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

Alternative 10: 2.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fmod(exp(x), ((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(exp(x), ((x * x) * (-0.25d0))) * (1.0d0 - x)
end function

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], 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 9.3%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 8.7

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

4. Applied rewrites 8.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. sub-flip-reverse N/A

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

   3. lower--.f64 7.7

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

7. Applied rewrites 7.7%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 2.7

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

10. Applied rewrites 2.7%

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

Reproduce

herbie shell --seed 2025134 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))