expfmod (used to be hard to sample)

Percentage Accurate: 8.7% → 40.3%

Time: 13.3s

Alternatives: 8

Speedup: 2.0×

• could not determine a ground truth

  1. x = -5.697311028931945e+234

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: 8.7% 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: 40.3% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

      4. sub-to-mult N/A

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

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

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

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

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

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

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

      8. lower-sinh.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-cosh.f64 8.7

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

   3. Applied rewrites 8.7%

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

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 2: 40.3% accurate, 0.5× speedup

Math

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

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 (* x x) -0.25 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((x * x), -0.25, 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(x * x), -0.25, 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[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.7%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.7

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

   3. Applied rewrites 8.7%

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

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 3: 39.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.3

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

   4. Applied rewrites 8.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

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

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

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

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

      20. lower-unsound--.f64 8.3

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

   6. Applied rewrites 8.3%

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

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 4: 39.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / 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[(x * x), $MachinePrecision] * -0.25 + 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[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 8.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.3

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

   4. Applied rewrites 8.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. div-flip N/A

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

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

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

      9. lower-unsound-/.f64 8.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 8.4

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 8.4

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

   6. Applied rewrites 8.4%

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

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 5: 39.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 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((x * x), -0.25, 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(x * x), -0.25, 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[(x * x), $MachinePrecision] * -0.25 + 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 8.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.3

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

   4. Applied rewrites 8.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 8.4

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 8.4

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 8.4

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

   6. Applied rewrites 8.4%

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

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

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 6: 38.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod t\_0\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 0.1)
     (* (fmod (exp x) t_0) (- 1.0 x))
     (/ (fmod 1.0 t_0) (exp x)))))

C

double code(double x) {
	double t_0 = fma((x * x), -0.25, 1.0);
	double tmp;
	if (x <= 0.1) {
		tmp = fmod(exp(x), t_0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, t_0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= 0.1)
		tmp = Float64(rem(exp(x), t_0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, t_0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.1], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 7.4

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

   4. Applied rewrites 7.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 7.4

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

   7. Applied rewrites 7.4%

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

   9. Applied rewrites 7.4%

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

   if 0.10000000000000001 < x

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

      1. 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 35.4

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

   7. Applied rewrites 35.4%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.4

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

   9. Applied rewrites 35.4%

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

Alternative 7: 35.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 35.4%

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

5. Taylor expanded in x around 0

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

   1. 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 35.4

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

7. Applied rewrites 35.4%

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
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 35.4

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

9. Applied rewrites 35.4%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 35.4%

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

5. Taylor expanded in x around 0

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

   1. 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 35.4

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

7. Applied rewrites 35.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 35.4

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

9. Applied rewrites 35.4%

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

10. Taylor expanded in x around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 4.6

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

12. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2025154 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))