expfmod (used to be hard to sample)

Percentage Accurate: 12.5% → 97.1%

Time: 32.7s

Alternatives: 15

Speedup: 3.5×

• could not determine a ground truth

  1. x = -5.507213223010669e-28

Specification

Math

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

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: 12.5% accurate, 1.0× speedup

Math

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

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: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := \left(\left(e^{x}\right) \bmod t\_0\right)\\ t_2 := e^{-x}\\ t_3 := t\_1 \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod t\_0\right) \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (cos x)))
       (t_1 (fmod (exp x) t_0))
       (t_2 (exp (- x)))
       (t_3 (* t_1 t_2)))
  (if (<= t_3 0)
    (*
     (fmod
      (*
       (/
        (- (pow 1 3) (pow (/ -1 x) 3))
        (+ (* 1 1) (+ (exp (* (log (* x x)) -1)) (/ -1 x))))
       x)
      t_0)
     t_2)
    (if (<= t_3 2)
      (/ 1 (/ (exp x) t_1))
      (* (fmod 1 (sqrt 1)) (- 1 x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = fmod(exp(x), t_0);
	double t_2 = exp(-x);
	double t_3 = t_1 * t_2;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = fmod((((pow(1.0, 3.0) - pow((-1.0 / x), 3.0)) / ((1.0 * 1.0) + (exp((log((x * x)) * -1.0)) + (-1.0 / x)))) * x), t_0) * t_2;
	} else if (t_3 <= 2.0) {
		tmp = 1.0 / (exp(x) / t_1);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * (1.0 - 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 = sqrt(cos(x))
    t_1 = mod(exp(x), t_0)
    t_2 = exp(-x)
    t_3 = t_1 * t_2
    if (t_3 <= 0.0d0) then
        tmp = mod(((((1.0d0 ** 3.0d0) - (((-1.0d0) / x) ** 3.0d0)) / ((1.0d0 * 1.0d0) + (exp((log((x * x)) * (-1.0d0))) + ((-1.0d0) / x)))) * x), t_0) * t_2
    else if (t_3 <= 2.0d0) then
        tmp = 1.0d0 / (exp(x) / t_1)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.sqrt(math.cos(x))
	t_1 = math.fmod(math.exp(x), t_0)
	t_2 = math.exp(-x)
	t_3 = t_1 * t_2
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.fmod((((math.pow(1.0, 3.0) - math.pow((-1.0 / x), 3.0)) / ((1.0 * 1.0) + (math.exp((math.log((x * x)) * -1.0)) + (-1.0 / x)))) * x), t_0) * t_2
	elif t_3 <= 2.0:
		tmp = 1.0 / (math.exp(x) / t_1)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * (1.0 - x)
	return tmp

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = rem(exp(x), t_0)
	t_2 = exp(Float64(-x))
	t_3 = Float64(t_1 * t_2)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(rem(Float64(Float64(Float64((1.0 ^ 3.0) - (Float64(-1.0 / x) ^ 3.0)) / Float64(Float64(1.0 * 1.0) + Float64(exp(Float64(log(Float64(x * x)) * -1.0)) + Float64(-1.0 / x)))) * x), t_0) * t_2);
	elseif (t_3 <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / t_1));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0], N[(N[With[{TMP1 = N[(N[(N[(N[Power[1, 3], $MachinePrecision] - N[Power[N[(-1 / x), $MachinePrecision], 3], $MachinePrecision]), $MachinePrecision] / N[(N[(1 * 1), $MachinePrecision] + N[(N[Exp[N[(N[Log[N[(x * x), $MachinePrecision]], $MachinePrecision] * -1), $MachinePrecision]], $MachinePrecision] + N[(-1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 2], N[(1 / N[(N[Exp[x], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1, TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 - 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 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 91.7%

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

   6. Applied rewrites 91.7%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. flip3-- N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      14. lower-unsound-*.f32 N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      15. lower-*.f32 N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + \left(\mathsf{neg}\left(1 \cdot \frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      17. *-lft-identity N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      4. frac-times N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\frac{-1 \cdot -1}{x \cdot x} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{x \cdot x} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      6. inv-pow N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left({\left(x \cdot x\right)}^{-1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      7. pow-to-exp N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      8. lower-unsound-exp.f64 N/A

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      10. lower-unsound-log.f64 N/A

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      11. lower-*.f64 92.2%

          \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 92.2%

       \[\leadsto \left(\left(\frac{{1}^{3} - {\left(\frac{-1}{x}\right)}^{3}}{1 \cdot 1 + \left(e^{\log \left(x \cdot x\right) \cdot -1} + \frac{-1}{x}\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

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

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. div-flip N/A

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

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

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

      6. lower-unsound-/.f64 12.6%

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

   5. Applied rewrites 12.6%

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

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

   1. Initial program 12.5%

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

   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 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 83.6%

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

   10. Applied rewrites 83.6%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       4. sub-flip-reverse N/A

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

       5. lower--.f64 83.6%

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

   12. Applied rewrites 83.6%

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

Alternative 2: 95.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (cos x))) (t_1 (sqrt (exp (+ x x)))))
  (if (<= (* (fmod (exp x) t_0) (exp (- x))) 2)
    (* (fmod t_1 t_0) (/ 1 t_1))
    (* (fmod 1 (sqrt 1)) (- 1 x)))))

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 12.5%

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

   5. Applied rewrites 12.5%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 12.5%

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

   7. Applied rewrites 12.5%

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

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

   1. Initial program 12.5%

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

   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 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 83.6%

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

   10. Applied rewrites 83.6%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       4. sub-flip-reverse N/A

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

       5. lower--.f64 83.6%

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

   12. Applied rewrites 83.6%

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

Alternative 3: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6530219459687219}{562949953421312}:\\ \;\;\;\;\left(\left(\sqrt{e^{x + x}}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 6530219459687219/562949953421312)
  (* (fmod (sqrt (exp (+ x x))) (sqrt (cos x))) (exp (- x)))
  (* (fmod 1 (sqrt 1)) (- 1 x))))

C

double code(double x) {
	double tmp;
	if (x <= 11.6) {
		tmp = fmod(sqrt(exp((x + x))), sqrt(cos(x))) * exp(-x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * (1.0 - x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 11.6d0) then
        tmp = mod(sqrt(exp((x + x))), sqrt(cos(x))) * exp(-x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 11.6:
		tmp = math.fmod(math.sqrt(math.exp((x + x))), math.sqrt(math.cos(x))) * math.exp(-x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 11.6)
		tmp = Float64(rem(sqrt(exp(Float64(x + x))), sqrt(cos(x))) * exp(Float64(-x)));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6530219459687219/562949953421312], N[(N[With[{TMP1 = N[Sqrt[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1, TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 11.6

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   if 11.6 < x

   1. Initial program 12.5%

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

   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 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 83.6%

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

   10. Applied rewrites 83.6%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       4. sub-flip-reverse N/A

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

       5. lower--.f64 83.6%

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

   12. Applied rewrites 83.6%

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

Alternative 4: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6530219459687219}{562949953421312}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 6530219459687219/562949953421312)
  (/ (fmod (exp x) (sqrt (cos x))) (exp x))
  (* (fmod 1 (sqrt 1)) (- 1 x))))

C

double code(double x) {
	double tmp;
	if (x <= 11.6) {
		tmp = fmod(exp(x), sqrt(cos(x))) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * (1.0 - x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 11.6d0) then
        tmp = mod(exp(x), sqrt(cos(x))) / exp(x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 11.6:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 11.6)
		tmp = Float64(rem(exp(x), sqrt(cos(x))) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6530219459687219/562949953421312], 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], N[(N[With[{TMP1 = 1, TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 11.6

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 12.6%

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

   5. Applied rewrites 12.6%

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

   if 11.6 < x

   1. Initial program 12.5%

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

   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 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 83.6%

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

   10. Applied rewrites 83.6%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       4. sub-flip-reverse N/A

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

       5. lower--.f64 83.6%

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

   12. Applied rewrites 83.6%

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

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6530219459687219}{562949953421312}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 6530219459687219/562949953421312)
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x)))
  (* (fmod 1 (sqrt 1)) (- 1 x))))

C

double code(double x) {
	double tmp;
	if (x <= 11.6) {
		tmp = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * (1.0 - x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 11.6d0) then
        tmp = mod(exp(x), sqrt(cos(x))) * exp(-x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * (1.0d0 - x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 11.6:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 11.6)
		tmp = Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6530219459687219/562949953421312], 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], N[(N[With[{TMP1 = 1, TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 11.6

   1. Initial program 12.5%

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

   if 11.6 < x

   1. Initial program 12.5%

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

   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 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 83.6%

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

   10. Applied rewrites 83.6%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

       4. sub-flip-reverse N/A

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

       5. lower--.f64 83.6%

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

   12. Applied rewrites 83.6%

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

Alternative 6: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{4} \cdot \left(x \cdot x\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq \frac{6530219459687219}{1125899906842624}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(1 + \frac{1}{t\_0}\right) \cdot t\_0\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + 2 \cdot x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* -1/4 (* x x))) (t_1 (exp (- x))))
  (if (<= x 6530219459687219/1125899906842624)
    (* (fmod (exp x) (* (+ 1 (/ 1 t_0)) t_0)) t_1)
    (* (fmod (sqrt (+ 1 (* 2 x))) (sqrt (cos x))) t_1))))

C

double code(double x) {
	double t_0 = -0.25 * (x * x);
	double t_1 = exp(-x);
	double tmp;
	if (x <= 5.8) {
		tmp = fmod(exp(x), ((1.0 + (1.0 / t_0)) * t_0)) * t_1;
	} else {
		tmp = fmod(sqrt((1.0 + (2.0 * x))), sqrt(cos(x))) * 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 = (-0.25d0) * (x * x)
    t_1 = exp(-x)
    if (x <= 5.8d0) then
        tmp = mod(exp(x), ((1.0d0 + (1.0d0 / t_0)) * t_0)) * t_1
    else
        tmp = mod(sqrt((1.0d0 + (2.0d0 * x))), sqrt(cos(x))) * t_1
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = -0.25 * (x * x)
	t_1 = math.exp(-x)
	tmp = 0
	if x <= 5.8:
		tmp = math.fmod(math.exp(x), ((1.0 + (1.0 / t_0)) * t_0)) * t_1
	else:
		tmp = math.fmod(math.sqrt((1.0 + (2.0 * x))), math.sqrt(math.cos(x))) * t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.25 * Float64(x * x))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64(rem(exp(x), Float64(Float64(1.0 + Float64(1.0 / t_0)) * t_0)) * t_1);
	else
		tmp = Float64(rem(sqrt(Float64(1.0 + Float64(2.0 * x))), sqrt(cos(x))) * t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1/4 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 6530219459687219/1125899906842624], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(1 + N[(1 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = N[Sqrt[N[(1 + N[(2 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999998

   1. Initial program 12.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 11.7%

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

   4. Applied rewrites 11.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

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

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

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

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

      6. lower-unsound-/.f64 11.7%

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 11.7%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 11.7%

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

   6. Applied rewrites 11.7%

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

   if 5.7999999999999998 < x

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 91.2%

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

   6. Applied rewrites 91.2%

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

Alternative 7: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6530219459687219}{1125899906842624}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot \left(x \cdot x\right) - -1\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + 2 \cdot x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 6530219459687219/1125899906842624)
  (/ (fmod (exp x) (- (* -1/4 (* x x)) -1)) (exp x))
  (* (fmod (sqrt (+ 1 (* 2 x))) (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8) {
		tmp = fmod(exp(x), ((-0.25 * (x * x)) - -1.0)) / exp(x);
	} else {
		tmp = fmod(sqrt((1.0 + (2.0 * x))), sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.8d0) then
        tmp = mod(exp(x), (((-0.25d0) * (x * x)) - (-1.0d0))) / exp(x)
    else
        tmp = mod(sqrt((1.0d0 + (2.0d0 * x))), sqrt(cos(x))) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 5.8:
		tmp = math.fmod(math.exp(x), ((-0.25 * (x * x)) - -1.0)) / math.exp(x)
	else:
		tmp = math.fmod(math.sqrt((1.0 + (2.0 * x))), math.sqrt(math.cos(x))) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64(rem(exp(x), Float64(Float64(-0.25 * Float64(x * x)) - -1.0)) / exp(x));
	else
		tmp = Float64(rem(sqrt(Float64(1.0 + Float64(2.0 * x))), sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6530219459687219/1125899906842624], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-1/4 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[Sqrt[N[(1 + N[(2 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999998

   1. Initial program 12.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 11.7%

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

   4. Applied rewrites 11.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot \left(x \cdot x\right) - -1\right)\right)}{\mathsf{Rewrite<=}\left(lift-exp.f64, \left(e^{x}\right)\right)} \]

   6. Applied rewrites 11.8%

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

   if 5.7999999999999998 < x

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 91.2%

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

   6. Applied rewrites 91.2%

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

Alternative 8: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6530219459687219}{1125899906842624}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot \left(x \cdot x\right) - -1\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 6530219459687219/1125899906842624)
  (/ (fmod (exp x) (- (* -1/4 (* x x)) -1)) (exp x))
  (* (fmod (+ 1 x) (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8) {
		tmp = fmod(exp(x), ((-0.25 * (x * x)) - -1.0)) / exp(x);
	} else {
		tmp = fmod((1.0 + x), sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.8d0) then
        tmp = mod(exp(x), (((-0.25d0) * (x * x)) - (-1.0d0))) / exp(x)
    else
        tmp = mod((1.0d0 + x), sqrt(cos(x))) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 5.8:
		tmp = math.fmod(math.exp(x), ((-0.25 * (x * x)) - -1.0)) / math.exp(x)
	else:
		tmp = math.fmod((1.0 + x), math.sqrt(math.cos(x))) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64(rem(exp(x), Float64(Float64(-0.25 * Float64(x * x)) - -1.0)) / exp(x));
	else
		tmp = Float64(rem(Float64(1.0 + x), sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6530219459687219/1125899906842624], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-1/4 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + 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. Split input into 2 regimes

2. if x < 5.7999999999999998

   1. Initial program 12.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 11.7%

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

   4. Applied rewrites 11.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot \left(x \cdot x\right) - -1\right)\right)}{\mathsf{Rewrite<=}\left(lift-exp.f64, \left(e^{x}\right)\right)} \]

   6. Applied rewrites 11.8%

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

   if 5.7999999999999998 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

Alternative 9: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq \frac{6530219459687219}{1125899906842624}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1}{4} \cdot \left(x \cdot x\right) - -1\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (exp (- x))))
  (if (<= x 6530219459687219/1125899906842624)
    (* (fmod (exp x) (- (* -1/4 (* x x)) -1)) t_0)
    (* (fmod (+ 1 x) (sqrt (cos x))) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 5.8) {
		tmp = fmod(exp(x), ((-0.25 * (x * x)) - -1.0)) * t_0;
	} else {
		tmp = fmod((1.0 + x), sqrt(cos(x))) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= 5.8d0) then
        tmp = mod(exp(x), (((-0.25d0) * (x * x)) - (-1.0d0))) * t_0
    else
        tmp = mod((1.0d0 + x), sqrt(cos(x))) * t_0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= 5.8:
		tmp = math.fmod(math.exp(x), ((-0.25 * (x * x)) - -1.0)) * t_0
	else:
		tmp = math.fmod((1.0 + x), math.sqrt(math.cos(x))) * t_0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64(rem(exp(x), Float64(Float64(-0.25 * Float64(x * x)) - -1.0)) * t_0);
	else
		tmp = Float64(rem(Float64(1.0 + x), sqrt(cos(x))) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 6530219459687219/1125899906842624], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-1/4 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999998

   1. Initial program 12.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 11.7%

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

   4. Applied rewrites 11.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 11.7%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 11.7%

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

   6. Applied rewrites 11.7%

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

   if 5.7999999999999998 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

Alternative 10: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\\ t_1 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(t\_0 \bmod t\_1\right) \cdot \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod t\_1\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (+ 1 (* x (+ 1 (* x (+ 1/2 (* 1/6 x)))))))
       (t_1 (sqrt (cos x))))
  (if (<= x 1)
    (* (fmod t_0 t_1) (/ 1 t_0))
    (* (fmod (+ 1 x) t_1) (exp (- x))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * (1.0 + (x * (0.5 + (0.16666666666666666 * x)))));
	double t_1 = sqrt(cos(x));
	double tmp;
	if (x <= 1.0) {
		tmp = fmod(t_0, t_1) * (1.0 / t_0);
	} else {
		tmp = fmod((1.0 + x), t_1) * 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) :: tmp
    t_0 = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (0.16666666666666666d0 * x)))))
    t_1 = sqrt(cos(x))
    if (x <= 1.0d0) then
        tmp = mod(t_0, t_1) * (1.0d0 / t_0)
    else
        tmp = mod((1.0d0 + x), t_1) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = 1.0 + (x * (1.0 + (x * (0.5 + (0.16666666666666666 * x)))))
	t_1 = math.sqrt(math.cos(x))
	tmp = 0
	if x <= 1.0:
		tmp = math.fmod(t_0, t_1) * (1.0 / t_0)
	else:
		tmp = math.fmod((1.0 + x), t_1) * math.exp(-x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(0.16666666666666666 * x))))))
	t_1 = sqrt(cos(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(rem(t_0, t_1) * Float64(1.0 / t_0));
	else
		tmp = Float64(rem(Float64(1.0 + x), t_1) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1 + N[(x * N[(1 + N[(x * N[(1/2 + N[(1/6 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1], N[(N[With[{TMP1 = t$95$0, TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = t$95$1}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 36.9%

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

   6. Applied rewrites 36.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 12.7%

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

   9. Applied rewrites 12.7%

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

   if 1 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

Alternative 11: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\\ \mathbf{if}\;x \leq \frac{5854679515581645}{4503599627370496}:\\ \;\;\;\;\left(t\_0 \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (+ 1 (* x (+ 1 (* x (+ 1/2 (* 1/6 x))))))))
  (if (<= x 5854679515581645/4503599627370496)
    (* (fmod t_0 (sqrt (cos x))) (/ 1 t_0))
    (* (fmod (+ 1 x) (sqrt 1)) (exp (- x))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * (1.0 + (x * (0.5 + (0.16666666666666666 * x)))));
	double tmp;
	if (x <= 1.3) {
		tmp = fmod(t_0, sqrt(cos(x))) * (1.0 / t_0);
	} else {
		tmp = fmod((1.0 + x), sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (0.16666666666666666d0 * x)))))
    if (x <= 1.3d0) then
        tmp = mod(t_0, sqrt(cos(x))) * (1.0d0 / t_0)
    else
        tmp = mod((1.0d0 + x), sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = 1.0 + (x * (1.0 + (x * (0.5 + (0.16666666666666666 * x)))))
	tmp = 0
	if x <= 1.3:
		tmp = math.fmod(t_0, math.sqrt(math.cos(x))) * (1.0 / t_0)
	else:
		tmp = math.fmod((1.0 + x), math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(0.16666666666666666 * x))))))
	tmp = 0.0
	if (x <= 1.3)
		tmp = Float64(rem(t_0, sqrt(cos(x))) * Float64(1.0 / t_0));
	else
		tmp = Float64(rem(Float64(1.0 + x), sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1 + N[(x * N[(1 + N[(x * N[(1/2 + N[(1/6 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5854679515581645/4503599627370496], N[(N[With[{TMP1 = t$95$0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 36.9%

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

   6. Applied rewrites 36.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 12.7%

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

   9. Applied rewrites 12.7%

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

   if 1.3 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 91.3%

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

Alternative 12: 94.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{if}\;x \leq \frac{5404319552844595}{9007199254740992}:\\ \;\;\;\;\left(t\_0 \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (+ 1 (* x (+ 1 (* 1/2 x))))))
  (if (<= x 5404319552844595/9007199254740992)
    (* (fmod t_0 (sqrt (cos x))) (/ 1 t_0))
    (* (fmod (+ 1 x) (sqrt 1)) (exp (- x))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * (1.0 + (0.5 * x)));
	double tmp;
	if (x <= 0.6) {
		tmp = fmod(t_0, sqrt(cos(x))) * (1.0 / t_0);
	} else {
		tmp = fmod((1.0 + x), sqrt(1.0)) * exp(-x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (1.0d0 + (0.5d0 * x)))
    if (x <= 0.6d0) then
        tmp = mod(t_0, sqrt(cos(x))) * (1.0d0 / t_0)
    else
        tmp = mod((1.0d0 + x), sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = 1.0 + (x * (1.0 + (0.5 * x)))
	tmp = 0
	if x <= 0.6:
		tmp = math.fmod(t_0, math.sqrt(math.cos(x))) * (1.0 / t_0)
	else:
		tmp = math.fmod((1.0 + x), math.sqrt(1.0)) * math.exp(-x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * Float64(1.0 + Float64(0.5 * x))))
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(rem(t_0, sqrt(cos(x))) * Float64(1.0 / t_0));
	else
		tmp = Float64(rem(Float64(1.0 + x), sqrt(1.0)) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1 + N[(x * N[(1 + N[(1/2 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5404319552844595/9007199254740992], N[(N[With[{TMP1 = t$95$0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.59999999999999998

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.9%

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

   6. Applied rewrites 49.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 13.1%

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

   9. Applied rewrites 13.1%

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

   if 0.59999999999999998 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 91.3%

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

Alternative 13: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{5404319552844595}{9007199254740992}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{1}\right)\right) \cdot e^{-x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 5404319552844595/9007199254740992)
  (* (fmod (+ 1 x) (sqrt (cos x))) (/ 1 (+ 1 x)))
  (* (fmod (+ 1 x) (sqrt 1)) (exp (- x)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = mod((1.0d0 + x), sqrt(cos(x))) * (1.0d0 / (1.0d0 + x))
    else
        tmp = mod((1.0d0 + x), sqrt(1.0d0)) * exp(-x)
    end if
    code = tmp
end function

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5404319552844595/9007199254740992], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 / N[(1 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.59999999999999998

   1. Initial program 12.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. lower-/.f64 12.5%

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

   3. Applied rewrites 12.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 91.4%

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

   6. Applied rewrites 91.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 15.0%

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

   9. Applied rewrites 15.0%

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

   if 0.59999999999999998 < x

   1. Initial program 12.5%

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

   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 91.4%

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 91.3%

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

Alternative 14: 91.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (fmod (+ 1 x) (sqrt 1)) (exp (- x))))

C

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(1 + x), $MachinePrecision], TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.5%

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

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 91.4%

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

4. Applied rewrites 91.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 91.3%

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

Alternative 15: 83.6% accurate, 3.5× speedup

Math

\[\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 - x\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (* (fmod 1 (sqrt 1)) (- 1 x)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = 1, TMP2 = N[Sqrt[1], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.5%

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

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 84.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 83.6%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 83.6%

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

10. Applied rewrites 83.6%

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

    1. lift-+.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. mul-1-neg N/A

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

    4. sub-flip-reverse N/A

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

    5. lower--.f64 83.6%

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

12. Applied rewrites 83.6%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))