expfmod (used to be hard to sample)

Percentage Accurate: 8.8% → 97.3%

Time: 15.9s

Alternatives: 12

Speedup: 1.4×

• could not determine a ground truth

  1. x = -3.953835256522836e+56

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: 8.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := 1 + -0.25 \cdot {x}^{2}\\ t_2 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\cosh x \cdot \sinh x, {\sinh x}^{-2}, 1\right) \cdot \sinh x\right) \bmod t\_2\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod t\_1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(2 \cdot \frac{\left(\left(e^{x}\right) \bmod t\_2\right)}{e^{x} - \frac{1}{e^{x}}}\right) \cdot \frac{1}{\frac{e^{x}}{\sinh x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (+ 1.0 (* -0.25 (pow x 2.0))))
        (t_2 (sqrt (cos x))))
   (if (<= x -2e-20)
     (*
      (fmod
       (* (fma (* (cosh x) (sinh x)) (pow (sinh x) -2.0) 1.0) (sinh x))
       t_2)
      t_0)
     (if (<= x 1.1e-16)
       (*
        (fmod
         (*
          (+
           1.0
           (/
            (/ (- (exp (+ x x)) (exp (- (- x) x))) (* 4.0 (sinh x)))
            (sinh x)))
          (sinh x))
         t_1)
        t_0)
       (if (<= x 50.0)
         (*
          (* 2.0 (/ (fmod (exp x) t_2) (- (exp x) (/ 1.0 (exp x)))))
          (/ 1.0 (/ (exp x) (sinh x))))
         (* (fmod 1.0 t_1) t_0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = 1.0 + (-0.25 * pow(x, 2.0));
	double t_2 = sqrt(cos(x));
	double tmp;
	if (x <= -2e-20) {
		tmp = fmod((fma((cosh(x) * sinh(x)), pow(sinh(x), -2.0), 1.0) * sinh(x)), t_2) * t_0;
	} else if (x <= 1.1e-16) {
		tmp = fmod(((1.0 + (((exp((x + x)) - exp((-x - x))) / (4.0 * sinh(x))) / sinh(x))) * sinh(x)), t_1) * t_0;
	} else if (x <= 50.0) {
		tmp = (2.0 * (fmod(exp(x), t_2) / (exp(x) - (1.0 / exp(x))))) * (1.0 / (exp(x) / sinh(x)));
	} else {
		tmp = fmod(1.0, t_1) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))
	t_2 = sqrt(cos(x))
	tmp = 0.0
	if (x <= -2e-20)
		tmp = Float64(rem(Float64(fma(Float64(cosh(x) * sinh(x)), (sinh(x) ^ -2.0), 1.0) * sinh(x)), t_2) * t_0);
	elseif (x <= 1.1e-16)
		tmp = Float64(rem(Float64(Float64(1.0 + Float64(Float64(Float64(exp(Float64(x + x)) - exp(Float64(Float64(-x) - x))) / Float64(4.0 * sinh(x))) / sinh(x))) * sinh(x)), t_1) * t_0);
	elseif (x <= 50.0)
		tmp = Float64(Float64(2.0 * Float64(rem(exp(x), t_2) / Float64(exp(x) - Float64(1.0 / exp(x))))) * Float64(1.0 / Float64(exp(x) / sinh(x))));
	else
		tmp = Float64(rem(1.0, t_1) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-20], N[(N[With[{TMP1 = N[(N[(N[(N[Cosh[x], $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sinh[x], $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$2}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.1e-16], N[(N[With[{TMP1 = N[(N[(1.0 + N[(N[(N[(N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision] - N[Exp[N[((-x) - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 50.0], N[(N[(2.0 * N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$2}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.99999999999999989e-20

   1. Initial program 8.8%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-sinh.f64 N/A

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

      10. lower-sinh.f64 8.7

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

   3. Applied rewrites 8.7%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-def N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. div-add N/A

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

      7. flip-+ N/A

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

      8. lower-unsound--.f32 N/A

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

      9. lower--.f32 N/A

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

      10. div-sub N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 63.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

   7. Applied rewrites 6.8%

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

   if -1.99999999999999989e-20 < x < 1.1e-16

   1. Initial program 8.8%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-sinh.f64 N/A

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

      10. lower-sinh.f64 8.7

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

   3. Applied rewrites 8.7%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-def N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. div-add N/A

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

      7. flip-+ N/A

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

      8. lower-unsound--.f32 N/A

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

      9. lower--.f32 N/A

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

      10. div-sub N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l/ N/A

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.5

         \[\leadsto \left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 63.5%

       \[\leadsto \left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]

   if 1.1e-16 < x < 50

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\cosh x + \sinh x}} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\cosh x} + \sinh x} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\cosh x + \color{blue}{\sinh x}} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\sinh x + \cosh x}} \]

      12. sum-to-mult-rev N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\left(1 + \frac{\cosh x}{\sinh x}\right) \cdot \sinh x}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\left(1 + \color{blue}{\frac{\cosh x}{\sinh x}}\right) \cdot \sinh x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\left(1 + \frac{\cosh x}{\sinh x}\right)} \cdot \sinh x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\sinh x \cdot \left(1 + \frac{\cosh x}{\sinh x}\right)}} \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\sinh x} \cdot \frac{1}{1 + \frac{\cosh x}{\sinh x}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\sinh x} \cdot \frac{1}{1 + \frac{\cosh x}{\sinh x}}} \]

   6. Applied rewrites 7.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fmod.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 5.0

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

   9. Applied rewrites 5.0%

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

   if 50 < x

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.7%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.7

         \[\leadsto \left(1 \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 34.7%

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

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := 1 + -0.25 \cdot {x}^{2}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod t\_3\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 0.5:\\ \;\;\;\;\left(2 \cdot \frac{t\_0}{e^{x} - \frac{1}{e^{x}}}\right) \cdot \frac{1}{\frac{e^{x}}{\sinh x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - -1\right) \bmod t\_3\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1))
        (t_3 (+ 1.0 (* -0.25 (pow x 2.0)))))
   (if (<= t_2 0.0)
     (*
      (fmod
       (*
        (+
         1.0
         (/ (/ (- (exp (+ x x)) (exp (- (- x) x))) (* 4.0 (sinh x))) (sinh x)))
        (sinh x))
       t_3)
      t_1)
     (if (<= t_2 0.5)
       (*
        (* 2.0 (/ t_0 (- (exp x) (/ 1.0 (exp x)))))
        (/ 1.0 (/ (exp x) (sinh x))))
       (/ (fmod (- x -1.0) t_3) (exp x))))))

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 t_3 = 1.0 + (-0.25 * pow(x, 2.0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = fmod(((1.0 + (((exp((x + x)) - exp((-x - x))) / (4.0 * sinh(x))) / sinh(x))) * sinh(x)), t_3) * t_1;
	} else if (t_2 <= 0.5) {
		tmp = (2.0 * (t_0 / (exp(x) - (1.0 / exp(x))))) * (1.0 / (exp(x) / sinh(x)));
	} else {
		tmp = fmod((x - -1.0), t_3) / 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    t_1 = exp(-x)
    t_2 = t_0 * t_1
    t_3 = 1.0d0 + ((-0.25d0) * (x ** 2.0d0))
    if (t_2 <= 0.0d0) then
        tmp = mod(((1.0d0 + (((exp((x + x)) - exp((-x - x))) / (4.0d0 * sinh(x))) / sinh(x))) * sinh(x)), t_3) * t_1
    else if (t_2 <= 0.5d0) then
        tmp = (2.0d0 * (t_0 / (exp(x) - (1.0d0 / exp(x))))) * (1.0d0 / (exp(x) / sinh(x)))
    else
        tmp = mod((x - (-1.0d0)), t_3) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.exp(-x)
	t_2 = t_0 * t_1
	t_3 = 1.0 + (-0.25 * math.pow(x, 2.0))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.fmod(((1.0 + (((math.exp((x + x)) - math.exp((-x - x))) / (4.0 * math.sinh(x))) / math.sinh(x))) * math.sinh(x)), t_3) * t_1
	elif t_2 <= 0.5:
		tmp = (2.0 * (t_0 / (math.exp(x) - (1.0 / math.exp(x))))) * (1.0 / (math.exp(x) / math.sinh(x)))
	else:
		tmp = math.fmod((x - -1.0), t_3) / math.exp(x)
	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)
	t_3 = Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(rem(Float64(Float64(1.0 + Float64(Float64(Float64(exp(Float64(x + x)) - exp(Float64(Float64(-x) - x))) / Float64(4.0 * sinh(x))) / sinh(x))) * sinh(x)), t_3) * t_1);
	elseif (t_2 <= 0.5)
		tmp = Float64(Float64(2.0 * Float64(t_0 / Float64(exp(x) - Float64(1.0 / exp(x))))) * Float64(1.0 / Float64(exp(x) / sinh(x))));
	else
		tmp = Float64(rem(Float64(x - -1.0), t_3) / exp(x));
	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]}, Block[{t$95$3 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[With[{TMP1 = N[(N[(1.0 + N[(N[(N[(N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision] - N[Exp[N[((-x) - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$3}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.5], N[(N[(2.0 * N[(t$95$0 / N[(N[Exp[x], $MachinePrecision] - N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = t$95$3}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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 8.8%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-sinh.f64 N/A

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

      10. lower-sinh.f64 8.7

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

   3. Applied rewrites 8.7%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-def N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. div-add N/A

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

      7. flip-+ N/A

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

      8. lower-unsound--.f32 N/A

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

      9. lower--.f32 N/A

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

      10. div-sub N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l/ N/A

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.5

         \[\leadsto \left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 63.5%

       \[\leadsto \left(\left(\left(1 + \frac{\frac{e^{x + x} - e^{\left(-x\right) - x}}{4 \cdot \sinh x}}{\sinh x}\right) \cdot \sinh x\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.5

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-exp.f64 N/A

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

      8. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\cosh x + \sinh x}} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\cosh x} + \sinh x} \]

      10. lift-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\cosh x + \color{blue}{\sinh x}} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\sinh x + \cosh x}} \]

      12. sum-to-mult-rev N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\left(1 + \frac{\cosh x}{\sinh x}\right) \cdot \sinh x}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\left(1 + \color{blue}{\frac{\cosh x}{\sinh x}}\right) \cdot \sinh x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\left(1 + \frac{\cosh x}{\sinh x}\right)} \cdot \sinh x} \]

      15. *-commutative N/A

          \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{\sinh x \cdot \left(1 + \frac{\cosh x}{\sinh x}\right)}} \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\sinh x} \cdot \frac{1}{1 + \frac{\cosh x}{\sinh x}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\sinh x} \cdot \frac{1}{1 + \frac{\cosh x}{\sinh x}}} \]

   6. Applied rewrites 7.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fmod.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 5.0

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

   9. Applied rewrites 5.0%

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

   if 0.5 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 37.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 37.8

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

   6. Applied rewrites 37.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 37.8

         \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right)}{e^{x}} \]

   9. Applied rewrites 37.8%

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

Alternative 3: 39.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot t\_1 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + \frac{\cosh x}{\sinh x}\right) \cdot \sinh x\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) t_0) t_1) 2.0)
     (* (fmod (* (+ 1.0 (/ (cosh x) (sinh x))) (sinh x)) t_0) t_1)
     (* (fmod 1.0 (+ 1.0 (* -0.25 (pow x 2.0)))) t_1))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), t_0) * t_1) <= 2.0) {
		tmp = fmod(((1.0 + (cosh(x) / sinh(x))) * sinh(x)), t_0) * t_1;
	} else {
		tmp = fmod(1.0, (1.0 + (-0.25 * pow(x, 2.0)))) * t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(cos(x))
    t_1 = exp(-x)
    if ((mod(exp(x), t_0) * t_1) <= 2.0d0) then
        tmp = mod(((1.0d0 + (cosh(x) / sinh(x))) * sinh(x)), t_0) * t_1
    else
        tmp = mod(1.0d0, (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) * t_1
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.sqrt(math.cos(x))
	t_1 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), t_0) * t_1) <= 2.0:
		tmp = math.fmod(((1.0 + (math.cosh(x) / math.sinh(x))) * math.sinh(x)), t_0) * t_1
	else:
		tmp = math.fmod(1.0, (1.0 + (-0.25 * math.pow(x, 2.0)))) * t_1
	return tmp

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), t_0) * t_1) <= 2.0)
		tmp = Float64(rem(Float64(Float64(1.0 + Float64(cosh(x) / sinh(x))) * sinh(x)), t_0) * t_1);
	else
		tmp = Float64(rem(1.0, Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[(N[(1.0 + N[(N[Cosh[x], $MachinePrecision] / N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $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 8.8%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-sinh.f64 N/A

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

      10. lower-sinh.f64 8.7

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

   3. Applied rewrites 8.7%

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

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

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.7%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.7

         \[\leadsto \left(1 \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 34.7%

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

Alternative 4: 39.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (exp (- x))))
   (if (<= (* t_0 t_1) 2.0)
     (/ t_0 (exp x))
     (* (fmod 1.0 (+ 1.0 (* -0.25 (pow x 2.0)))) t_1))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double tmp;
	if ((t_0 * t_1) <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, (1.0 + (-0.25 * pow(x, 2.0)))) * t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    t_1 = exp(-x)
    if ((t_0 * t_1) <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = mod(1.0d0, (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) * t_1
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.exp(-x)
	tmp = 0
	if (t_0 * t_1) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = math.fmod(1.0, (1.0 + (-0.25 * math.pow(x, 2.0)))) * t_1
	return tmp

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * t$95$1), $MachinePrecision], 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $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 8.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg 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 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.8

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

   3. Applied rewrites 8.8%

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

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

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.7%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.7

         \[\leadsto \left(1 \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 34.7%

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

Alternative 5: 39.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (/ 1.0 (/ (exp x) (fmod (exp x) (fma (* x x) -0.25 1.0))))
     (* (fmod 1.0 (+ 1.0 (* -0.25 (pow x 2.0)))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = 1.0 / (exp(x) / fmod(exp(x), fma((x * x), -0.25, 1.0)));
	} else {
		tmp = fmod(1.0, (1.0 + (-0.25 * pow(x, 2.0)))) * t_0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $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 8.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg 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 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.8

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

   3. Applied rewrites 8.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.5

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right)}{e^{x}} \]

   6. Applied rewrites 8.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 8.5

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 8.5

         \[\leadsto \frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{-0.25}, 1\right)\right)\right)}} \]

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 8.5

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

   8. Applied rewrites 8.5%

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

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

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.7%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.7

         \[\leadsto \left(1 \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot e^{-x} \]

   7. Applied rewrites 34.7%

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

Alternative 6: 37.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (* (+ 1.0 (/ 1.0 x)) x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(((1.0 + (1.0 / x)) * 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(((1.0d0 + (1.0d0 / x)) * x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(((1.0 + (1.0 / x)) * x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(Float64(Float64(1.0 + Float64(1.0 / x)) * x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 37.8

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

4. Applied rewrites 37.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. sum-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound-+.f64 N/A

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

   6. lower-unsound-/.f64 37.9

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

6. Applied rewrites 37.9%

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

Alternative 7: 37.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 37.8

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

4. Applied rewrites 37.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 37.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. add-flip N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 37.8

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

6. Applied rewrites 37.8%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-/.f64 37.9

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

8. Applied rewrites 37.9%

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

Alternative 8: 37.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fmod (- x -1.0) (sqrt (cos x))) (exp x)))

C

double code(double x) {
	return fmod((x - -1.0), 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((x - (-1.0d0)), sqrt(cos(x))) / exp(x)
end function

Python

def code(x):
	return math.fmod((x - -1.0), math.sqrt(math.cos(x))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(Float64(x - -1.0), sqrt(cos(x))) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 37.8

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

4. Applied rewrites 37.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 37.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. add-flip N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 37.8

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

6. Applied rewrites 37.8%

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

Alternative 9: 37.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x - -1\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (- x -1.0) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x)))

C

double code(double x) {
	return fmod((x - -1.0), (1.0 + (-0.25 * pow(x, 2.0)))) / 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((x - (-1.0d0)), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
end function

Python

def code(x):
	return math.fmod((x - -1.0), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(Float64(x - -1.0), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 37.8

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

4. Applied rewrites 37.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 37.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. add-flip N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 37.8

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

6. Applied rewrites 37.8%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 37.8

      \[\leadsto \frac{\left(\left(x - -1\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right)}{e^{x}} \]

9. Applied rewrites 37.8%

   \[\leadsto \frac{\left(\left(x - -1\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
10. Add Preprocessing

Alternative 10: 8.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg 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 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 8.8

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

3. Applied rewrites 8.8%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 8.5

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right)}{e^{x}} \]

6. Applied rewrites 8.5%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 8.5

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}}} \]

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 8.5

      \[\leadsto \frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{-0.25}, 1\right)\right)\right)}} \]

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 8.5

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

8. Applied rewrites 8.5%

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

Alternative 11: 8.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg 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 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 8.8

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

3. Applied rewrites 8.8%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 8.5

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{\color{blue}{2}}\right)\right)}{e^{x}} \]

6. Applied rewrites 8.5%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 8.5

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, \color{blue}{-0.25}, 1\right)\right)\right)}{e^{x}} \]

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 8.5

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

8. Applied rewrites 8.5%

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

Alternative 12: 6.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

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. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 6.5

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

4. Applied rewrites 6.5%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 6.5

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right) \]

7. Applied rewrites 6.5%

   \[\leadsto \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{-0.25 \cdot {x}^{2}}\right)\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 6.5

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left({x}^{2}, -0.25, 1\right)\right)\right) \]

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 6.5

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

9. Applied rewrites 6.5%

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

Reproduce

herbie shell --seed 2025155 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))