expfmod (used to be hard to sample)

Percentage Accurate: 8.9% → 40.1%

Time: 14.9s

Alternatives: 8

Speedup: 1.9×

• could not determine a ground truth

  1. x = -8.096068130955141e-142

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Initial Program: 8.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 40.1% accurate, 0.5× speedup

Math

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

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 8.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.1%

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

      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. *-commutative N/A

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

      12. lower-fma.f64 35.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

Alternative 2: 39.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / 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_] := 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 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], 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 8.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.5%

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

   6. Applied rewrites 8.5%

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

   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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.1%

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

      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. *-commutative N/A

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

      12. lower-fma.f64 35.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

Alternative 3: 39.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 35.1%

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

   10. Taylor expanded in x around inf

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

       1. lower-exp.f64 8.5%

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

   12. Applied rewrites 8.5%

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

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

   1. Initial program 8.9%

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

   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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.1%

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

      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. *-commutative N/A

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

      12. lower-fma.f64 35.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

Alternative 4: 38.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

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 35.1%

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

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 35.1%

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

7. Applied rewrites 35.1%

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
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. *-commutative N/A

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

   5. lower-fma.f64 35.1%

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 35.1%

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

9. Applied rewrites 35.1%

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

10. Taylor expanded in x around 0

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

    1. lower-+.f64 38.3%

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

12. Applied rewrites 38.3%

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

Alternative 5: 37.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 216:\\ \;\;\;\;\left(\left(\sqrt{e^{x + x}}\right) \bmod t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod t\_0\right)}{e^{x}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 216.0)
     (fmod (sqrt (exp (+ x 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 (x <= 216.0) {
		tmp = fmod(sqrt(exp((x + 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 (x <= 216.0)
		tmp = rem(sqrt(exp(Float64(x + 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[x, 216.0], N[With[{TMP1 = N[Sqrt[N[Exp[N[(x + x), $MachinePrecision]], $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 < 216

   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. *-commutative N/A

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

      5. lower-fma.f64 6.5%

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 6.5%

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

   9. Applied rewrites 6.5%

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

       1. lift-exp.f64 N/A

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

       2. exp-fabs N/A

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

       3. lift-exp.f64 N/A

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

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

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

       5. lift-exp.f64 N/A

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

       6. lift-exp.f64 N/A

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

       7. exp-sum N/A

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

       8. lift-+.f64 N/A

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

       9. lift-exp.f64 N/A

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

       10. lift-sqrt.f64 6.6%

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

   11. Applied rewrites 6.6%

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

   if 216 < 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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.1%

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

      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. *-commutative N/A

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

      12. lower-fma.f64 35.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

Alternative 6: 37.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\\ \mathbf{if}\;x \leq 216:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod t\_0\right)}{e^{x}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.25 1.0)))
   (if (<= x 216.0) (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 (x <= 216.0) {
		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(x * x), -0.25, 1.0)
	tmp = 0.0
	if (x <= 216.0)
		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[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, If[LessEqual[x, 216.0], 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 < 216

   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. *-commutative N/A

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

      5. lower-fma.f64 6.5%

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 6.5%

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

   9. Applied rewrites 6.5%

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

   if 216 < 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 35.1%

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

   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 35.1%

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

   7. Applied rewrites 35.1%

      \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
   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 35.1%

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

      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. *-commutative N/A

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

      12. lower-fma.f64 35.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 35.1%

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

   9. Applied rewrites 35.1%

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

Alternative 7: 6.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

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. *-commutative N/A

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

   5. lower-fma.f64 6.5%

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 6.5%

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

9. Applied rewrites 6.5%

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

Alternative 8: 4.7% accurate, 2.3× speedup

Math

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

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 8.9%

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

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 35.1%

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

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 35.1%

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

7. Applied rewrites 35.1%

   \[\leadsto \left(1 \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \cdot e^{-x} \]
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. *-commutative N/A

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

   5. lower-fma.f64 35.1%

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 35.1%

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

9. Applied rewrites 35.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
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 4.7%

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

12. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2025193 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))