expfmod (used to be hard to sample)

Percentage Accurate: 8.6% → 40.3%

Time: 13.6s

Alternatives: 8

Speedup: 2.2×

• could not determine a ground truth

  1. x = -1.655285515025782e+277

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: 8.6% 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: 40.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

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

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

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

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

      9. lower-unsound-/.f64 8.6%

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

   3. Applied rewrites 8.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

      6. lower-unsound-log.f64 8.6%

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

   5. Applied rewrites 8.6%

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

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

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 2: 40.3% accurate, 0.5× speedup

Math

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

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 (* -0.25 x) x 1.0)) (exp x)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 8.6%

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

   3. Applied rewrites 8.6%

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

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

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 3: 40.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[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[Exp[N[(N[Log[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] * -1.0), $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 8.6%

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

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

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

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

      9. lower-unsound-/.f64 8.6%

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

   3. Applied rewrites 8.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.3%

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

   6. Applied rewrites 8.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

   8. Applied rewrites 8.3%

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

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

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 4: 40.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

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

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

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

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

      9. lower-unsound-/.f64 8.6%

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

   3. Applied rewrites 8.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 8.3%

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

   6. Applied rewrites 8.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lower-/.f64 8.3%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 8.3%

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

   8. Applied rewrites 8.3%

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

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

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 5: 39.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\\ \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod t\_0\right)}{e^{x}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* -0.25 x) x 1.0)))
   (if (<= x 0.98)
     (* (fmod (exp x) t_0) (- 1.0 x))
     (/ (fmod 1.0 t_0) (exp x)))))

C

double code(double x) {
	double t_0 = fma((-0.25 * x), x, 1.0);
	double tmp;
	if (x <= 0.98) {
		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(-0.25 * x), x, 1.0)
	tmp = 0.0
	if (x <= 0.98)
		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[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.98], 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 < 0.97999999999999998

   1. Initial program 8.6%

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

   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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 7.4%

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

      11. lift-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{4} \cdot x, \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + -1 \cdot x\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{4} \cdot x, \color{blue}{x}, 1\right)\right)\right) \cdot \left(1 + \mathsf{Rewrite=>}\left(lift-*.f64, \left(-1 \cdot x\right)\right)\right) \]

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{4} \cdot x, \color{blue}{x}, 1\right)\right)\right) \cdot \mathsf{Rewrite=>}\left(lower--.f64, \left(1 - x\right)\right) \]

   9. Applied rewrites 7.4%

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

   if 0.97999999999999998 < x

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 6: 38.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod((1.0 + x), fma((x * x), -0.25, 1.0));
	} else {
		tmp = fmod(1.0, fma((-0.25 * x), x, 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 = rem(Float64(1.0 + x), fma(Float64(x * x), -0.25, 1.0));
	else
		tmp = Float64(rem(1.0, fma(Float64(-0.25 * x), x, 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[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[(N[With[{TMP1 = 1.0, TMP2 = N[(N[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

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

   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. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-neg-in N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-neg-out N/A

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

      21. mul-1-neg N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f32 N/A

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

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

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

      25. lower-fma.f64 N/A

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

   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. Taylor expanded in x around 0

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

       1. lower-+.f64 7.2%

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

   12. Applied rewrites 7.2%

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

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

   1. Initial program 8.6%

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

   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 35.5%

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

   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 35.5%

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

   7. Applied rewrites 35.5%

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

   9. Applied rewrites 35.5%

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

Alternative 7: 21.3% accurate, 2.2× speedup

Math

\[\left(\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fmod (+ 1.0 (* x (+ 1.0 (* 0.5 x)))) (fma (* x x) -0.25 1.0)))

C

double code(double x) {
	return fmod((1.0 + (x * (1.0 + (0.5 * x)))), fma((x * x), -0.25, 1.0));
}

Julia

function code(x)
	return rem(Float64(1.0 + Float64(x * Float64(1.0 + Float64(0.5 * x)))), fma(Float64(x * x), -0.25, 1.0))
end

Wolfram

code[x_] := N[With[{TMP1 = N[(1.0 + N[(x * N[(1.0 + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 8.6%

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

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. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. distribute-lft-neg-in N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

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

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

   18. mul-1-neg N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-neg-out N/A

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

   21. mul-1-neg N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f32 N/A

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

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

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

   25. lower-fma.f64 N/A

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

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. Taylor expanded in x around 0

    \[\leadsto \left(\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-*.f64 21.3%

       \[\leadsto \left(\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \]

12. Applied rewrites 21.3%

    \[\leadsto \left(\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 1\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 7.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := 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]

Derivation

1. Initial program 8.6%

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

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. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. distribute-lft-neg-in N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

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

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

   18. mul-1-neg N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-neg-out N/A

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

   21. mul-1-neg N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f32 N/A

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

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

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

   25. lower-fma.f64 N/A

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

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. Taylor expanded in x around 0

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

    1. lower-+.f64 7.2%

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

12. Applied rewrites 7.2%

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

Reproduce

herbie shell --seed 2025179 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))