expfmod (used to be hard to sample)

Percentage Accurate: 9.3% → 47.8%

Time: 14.9s

Alternatives: 14

Speedup: 2.2×

• could not determine a ground truth

  1. x = -1.949195577684668e-40

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: 47.8% accurate, 0.3× speedup

Math

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

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 1.0 (* (fma 0.5 x 1.0) (* -0.5 x))) t_1)
     (if (<= t_2 2.0)
       (* t_0 (/ 1.0 (exp x)))
       (* (fmod 1.0 (* (* x x) -0.25)) 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(1.0, (fma(0.5, x, 1.0) * (-0.5 * x))) * t_1;
	} else if (t_2 <= 2.0) {
		tmp = t_0 * (1.0 / exp(x));
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * 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(1.0, Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_1);
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 * Float64(1.0 / exp(x)));
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * 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 = 1.0, 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[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 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 5.4%

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

   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 5.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 5.4

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

   9. Applied rewrites 5.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

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

   1. Initial program 81.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 81.1

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

   3. Applied rewrites 81.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

Alternative 2: 47.8% accurate, 0.3× speedup

Math

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

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 1.0 (* (fma 0.5 x 1.0) (* -0.5 x))) t_0)
     (if (<= t_1 2.0) t_1 (* (fmod 1.0 (* (* x x) -0.25)) 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(1.0, (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * 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(1.0, Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_0);
	elseif (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * 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 = 1.0, 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], t$95$1, N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $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 5.4%

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

   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 5.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 5.4

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

   9. Applied rewrites 5.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

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

   1. Initial program 81.0%

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

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

   1. Initial program 0.0%

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

   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 98.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

Alternative 3: 46.1% accurate, 0.6× speedup

Math

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

FPCore

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

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(1.0, (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	} else {
		tmp = fmod((x - -1.0), ((((1.0 / (x * x)) - 0.25) * x) * x)) * t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.4%

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

   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 5.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 5.4

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

   9. Applied rewrites 5.4%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

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

   1. Initial program 15.3%

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

   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 81.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 81.8

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

   10. Applied rewrites 81.8%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. lower--.f64 90.4

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

   13. Applied rewrites 90.4%

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

Alternative 4: 46.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right) \cdot 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.6e-17)
   (/ (* (fmod (exp x) 1.0) 1.0) (exp x))
   (* (fmod 1.0 (* (fma 0.5 x 1.0) (* -0.5 x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5.6e-17) {
		tmp = (fmod(exp(x), 1.0) * 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, (fma(0.5, x, 1.0) * (-0.5 * x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.6e-17)
		tmp = Float64(Float64(rem(exp(x), 1.0) * 1.0) / exp(x));
	else
		tmp = Float64(rem(1.0, Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5.6e-17], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999998e-17

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 87.1

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

   6. Applied rewrites 87.1%

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

   if -5.5999999999999998e-17 < x

   1. Initial program 5.9%

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

   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 36.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 36.7

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

   7. Applied rewrites 36.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 36.8

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

   9. Applied rewrites 36.8%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 44.3

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

   12. Applied rewrites 44.3%

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

Alternative 5: 46.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \left(-0.5 \cdot x\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5.6e-17)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod 1.0 (* (fma 0.5 x 1.0) (* -0.5 x))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -5.6e-17) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(1.0, (fma(0.5, x, 1.0) * (-0.5 * x))) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5.6e-17)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(fma(0.5, x, 1.0) * Float64(-0.5 * x))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5.6e-17], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999998e-17

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.7%

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

   if -5.5999999999999998e-17 < x

   1. Initial program 5.9%

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

   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 36.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 36.7

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

   7. Applied rewrites 36.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. metadata-eval N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 36.8

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

   9. Applied rewrites 36.8%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 44.3

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

   12. Applied rewrites 44.3%

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

Alternative 6: 40.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod 1.0 (* (* x x) -0.25)) 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), 1.0) * t_0;
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = mod(exp(x), 1.0d0) * t_0
    else
        tmp = mod(1.0d0, ((x * x) * (-0.25d0))) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = math.fmod(math.exp(x), 1.0) * t_0
	else:
		tmp = math.fmod(1.0, ((x * x) * -0.25)) * 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), 1.0) * t_0);
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * 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 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25), $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 13.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 12.5%

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

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

   1. Initial program 0.0%

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

   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 98.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

Alternative 7: 39.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right) \bmod t\_0\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_0\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 0.92)
     (* (fmod (fma (fma 0.5 x 1.0) x 1.0) t_0) (- 1.0 x))
     (* (fmod 1.0 t_0) (exp (- x))))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= 0.92) {
		tmp = fmod(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, t_0) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= 0.92)
		tmp = Float64(rem(fma(fma(0.5, x, 1.0), x, 1.0), t_0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, t_0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.92], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, 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 x < 0.92000000000000004

   1. Initial program 13.4%

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

   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 5.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 5.7

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

   7. Applied rewrites 5.7%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-lft-identity N/A

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

      4. lower--.f64 5.8

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-fma.f64 11.5

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

   13. Applied rewrites 11.5%

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

   if 0.92000000000000004 < x

   1. Initial program 0.6%

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

   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 99.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

Alternative 8: 39.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod 1.0 (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -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 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 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 13.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 11.4

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

   7. Applied rewrites 11.4%

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

   if 0.880000000000000004 < x

   1. Initial program 0.6%

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

   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 99.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

Alternative 9: 39.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod 1.0 (fma (* x x) -0.25 1.0)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, 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] * -0.25 + 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 < 0.599999999999999978

   1. Initial program 13.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 11.1

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

   7. Applied rewrites 11.1%

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

   if 0.599999999999999978 < x

   1. Initial program 0.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 10: 39.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\left(x \cdot x\right) \cdot -0.25\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod 1.0 (* (* x x) -0.25)) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, ((x * x) * -0.25)) * exp(-x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(1.0d0, ((x * x) * (-0.25d0))) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(1.0, ((x * x) * -0.25)) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, Float64(Float64(x * x) * -0.25)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, 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] * -0.25), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 13.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 11.1

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

   7. Applied rewrites 11.1%

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

   if 0.599999999999999978 < x

   1. Initial program 0.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.3

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

   10. Applied rewrites 99.3%

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

Alternative 11: 39.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 13.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 12.4%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 11.1

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

   7. Applied rewrites 11.1%

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

   if 1 < x

   1. Initial program 0.6%

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

   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 99.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

Alternative 12: 38.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 13.4%

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

   2. Taylor expanded in x around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fmod.f64 N/A

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

      4. lift-exp.f64 10.0

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 10.0%

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

   if 1 < x

   1. Initial program 0.6%

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

   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 99.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

Alternative 13: 8.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (fmod (exp x) (sqrt 1.0))
   (* (fmod 1.0 (fma (* x x) -0.25 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), sqrt(1.0));
	} else {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = rem(exp(x), sqrt(1.0));
	else
		tmp = Float64(rem(1.0, fma(Float64(x * x), -0.25, 1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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 13.8%

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

   2. Taylor expanded in x around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fmod.f64 N/A

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

      4. lift-exp.f64 10.1

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

   4. Applied rewrites 10.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 10.1%

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

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

   1. Initial program 0.0%

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

   2. 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 98.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-lft-identity N/A

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

      4. lower--.f64 2.4

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 4.8%

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

Alternative 14: 5.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 35.4

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

7. Applied rewrites 35.4%

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

8. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. lower--.f64 4.6

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

10. Applied rewrites 4.6%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 5.6%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))