expfmod (used to be hard to sample)

Percentage Accurate: 8.9% → 47.8%

Time: 21.6s

Alternatives: 8

Speedup: 1.9×

• could not determine a ground truth

  1. x = -7.615422295738604e+66

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 8.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 47.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := \left(\left(e^{x}\right) \bmod t\_0\right)\\ t_2 := \log \cosh x\\ t_3 := t\_1 \cdot e^{-x}\\ t_4 := {\left(e^{1 - \frac{\log \sinh x}{t\_2}}\right)}^{t\_2}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\left(\left(\frac{\left(e^{\log t\_4 \cdot 3} + 1\right) \cdot \sinh x}{\mathsf{fma}\left(t\_4, t\_4 - 1, 1\right)}\right) \bmod t\_0\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{t\_1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x)))
        (t_1 (fmod (exp x) t_0))
        (t_2 (log (cosh x)))
        (t_3 (* t_1 (exp (- x))))
        (t_4 (pow (exp (- 1.0 (/ (log (sinh x)) t_2))) t_2)))
   (if (<= t_3 0.0)
     (fmod
      (/
       (* (+ (exp (* (log t_4) 3.0)) 1.0) (sinh x))
       (fma t_4 (- t_4 1.0) 1.0))
      t_0)
     (if (<= t_3 2.0)
       (/ t_1 (exp x))
       (/ (fmod 1.0 (fma (* -0.25 x) x 1.0)) (exp x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = fmod(exp(x), t_0);
	double t_2 = log(cosh(x));
	double t_3 = t_1 * exp(-x);
	double t_4 = pow(exp((1.0 - (log(sinh(x)) / t_2))), t_2);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = fmod((((exp((log(t_4) * 3.0)) + 1.0) * sinh(x)) / fma(t_4, (t_4 - 1.0), 1.0)), t_0);
	} else if (t_3 <= 2.0) {
		tmp = t_1 / exp(x);
	} else {
		tmp = fmod(1.0, fma((-0.25 * x), x, 1.0)) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = rem(exp(x), t_0)
	t_2 = log(cosh(x))
	t_3 = Float64(t_1 * exp(Float64(-x)))
	t_4 = exp(Float64(1.0 - Float64(log(sinh(x)) / t_2))) ^ t_2
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = rem(Float64(Float64(Float64(exp(Float64(log(t_4) * 3.0)) + 1.0) * sinh(x)) / fma(t_4, Float64(t_4 - 1.0), 1.0)), t_0);
	elseif (t_3 <= 2.0)
		tmp = Float64(t_1 / exp(x));
	else
		tmp = Float64(rem(1.0, fma(Float64(-0.25 * x), x, 1.0)) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Cosh[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Exp[N[(1.0 - N[(N[Log[N[Sinh[x], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[With[{TMP1 = N[(N[(N[(N[Exp[N[(N[Log[t$95$4], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * N[(t$95$4 - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$1 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.5

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

   4. Applied rewrites 6.5%

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

      1. lift-exp.f64 N/A

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

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

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

      3. lift-cosh.f64 N/A

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

      4. lift-sinh.f64 N/A

         \[\leadsto \left(\left(\cosh x + \sinh x\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

      5. +-commutative N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 4.2%

      \[\leadsto \left(\left(\frac{\left({\left(\frac{\cosh x}{\sinh x}\right)}^{3} + 1\right) \cdot \sinh x}{\mathsf{fma}\left(\frac{\cosh x}{\sinh x}, \frac{\cosh x}{\sinh x} - 1, 1\right)}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(\frac{\left({\left(\frac{\cosh x}{\sinh x}\right)}^{3} + 1\right) \cdot \sinh x}{\mathsf{fma}\left(\frac{\cosh x}{\sinh x}, \frac{\cosh x}{\sinh x} - 1, 1\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 2.7

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

   8. Applied rewrites 2.7%

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

      1. rem-exp-log N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. sub-to-mult N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 4.0

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

   10. Applied rewrites 4.0%

       \[\leadsto \left(\left(\frac{\left(e^{\log \left({\left(e^{1 - \frac{\log \sinh x}{\log \cosh x}}\right)}^{\log \cosh x}\right) \cdot 3} + 1\right) \cdot \sinh x}{\mathsf{fma}\left(\frac{\cosh x}{\sinh x}, \frac{\cosh x}{\sinh x} - 1, 1\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]
   11. Step-by-step derivation

       1. rem-exp-log N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. sub-to-mult N/A

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

       5. exp-prod N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-log.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-log.f64 4.5

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

   12. Applied rewrites 4.5%

       \[\leadsto \left(\left(\frac{\left(e^{\log \left({\left(e^{1 - \frac{\log \sinh x}{\log \cosh x}}\right)}^{\log \cosh x}\right) \cdot 3} + 1\right) \cdot \sinh x}{\mathsf{fma}\left({\left(e^{1 - \frac{\log \sinh x}{\log \cosh x}}\right)}^{\log \cosh x}, \frac{\cosh x}{\sinh x} - 1, 1\right)}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]
   13. Step-by-step derivation

       1. rem-exp-log N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. sub-to-mult N/A

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

       5. exp-prod N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-log.f64 N/A

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

       11. lower-log.f64 N/A

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

       12. lower-log.f64 11.8

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

   14. Applied rewrites 11.8%

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

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

   1. Initial program 8.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.9

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

   3. Applied rewrites 8.9%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 2: 39.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.9

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

   3. Applied rewrites 8.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. lift-cosh.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 8.8

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. add-to-fraction N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. lift-cosh.f64 N/A

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

      16. lift-sinh.f64 N/A

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

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

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

      18. lift-exp.f64 N/A

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

      19. lower-/.f64 9.0

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

   5. Applied rewrites 9.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. lift-cosh.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 8.8

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. add-to-fraction N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. lift-cosh.f64 N/A

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

      16. lift-sinh.f64 N/A

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

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

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

      18. lift-exp.f64 N/A

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

      19. lower-/.f64 9.0

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

   7. Applied rewrites 9.0%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 3: 39.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.9

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

   3. Applied rewrites 8.9%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 4: 39.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

   10. Taylor expanded in x around inf

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

       1. lower-exp.f64 8.5

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

   12. Applied rewrites 8.5%

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

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

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 5: 38.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 34.9%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 34.9

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

7. Applied rewrites 34.9%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 34.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 34.9

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

9. Applied rewrites 34.9%

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

10. Taylor expanded in x around 0

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

    1. lower-+.f64 38.0

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

12. Applied rewrites 38.0%

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

Alternative 6: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 6.5

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

   7. Applied rewrites 6.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 6.5

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

   9. Applied rewrites 6.5%

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

       1. lift-exp.f64 N/A

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

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

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

       3. lift-cosh.f64 N/A

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

       4. lift-sinh.f64 N/A

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

       5. add-flip N/A

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

       6. lift-cosh.f64 N/A

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

       7. cosh-neg-rev N/A

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

       8. lift-neg.f64 N/A

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

       9. lift-sinh.f64 N/A

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

       10. sinh-neg-rev N/A

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

       11. lift-neg.f64 N/A

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

       12. sinh---cosh-rev N/A

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

       13. exp-neg N/A

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

       14. lift-exp.f64 N/A

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

       15. lower-/.f64 6.5

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

   11. Applied rewrites 6.5%

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

   if 1 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 7: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 6.5

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

   7. Applied rewrites 6.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 6.5

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

   9. Applied rewrites 6.5%

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

   if 1 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.9

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 34.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

Alternative 8: 6.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

2. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 6.5

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

4. Applied rewrites 6.5%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 6.5

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

7. Applied rewrites 6.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 6.5

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

9. Applied rewrites 6.5%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))