x / (x^2 + 1)

Percentage Accurate: 76.9% → 100.0%

Time: 3.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

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 = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

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 = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot x\_m, \left|x\_m\right|, -1\right)} \cdot \mathsf{fma}\left(x\_m, x\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 20000000.0)
    (* (/ x_m (fma (* (* x_m x_m) x_m) (fabs x_m) -1.0)) (fma x_m x_m -1.0))
    (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = (x_m / fma(((x_m * x_m) * x_m), fabs(x_m), -1.0)) * fma(x_m, x_m, -1.0);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(Float64(x_m / fma(Float64(Float64(x_m * x_m) * x_m), abs(x_m), -1.0)) * fma(x_m, x_m, -1.0));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(N[(x$95$m / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 84.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x \cdot x + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x + 1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x} + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2}} + 1} \]

      5. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{{x}^{2} \cdot {x}^{2} - 1 \cdot 1}{{x}^{2} - 1}}} \]

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. sqr-abs-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-fabs.f64 72.6

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

   6. Applied rewrites 72.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. pow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \sqrt{\color{blue}{{x}^{2}}}, \left|x\right|, -1\right)} \cdot \mathsf{fma}\left(x, x, -1\right) \]

      4. sqrt-pow1 N/A

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

      5. metadata-eval N/A

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

      6. unpow1 61.8

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

   8. Applied rewrites 61.8%

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

   if 2e7 < x

   1. Initial program 57.6%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 100.0

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 100.0

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot x\_m, -1\right)} \cdot \mathsf{fma}\left(x\_m, x\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 20000000.0)
    (* (/ x_m (fma (* x_m x_m) (* x_m x_m) -1.0)) (fma x_m x_m -1.0))
    (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = (x_m / fma((x_m * x_m), (x_m * x_m), -1.0)) * fma(x_m, x_m, -1.0);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(Float64(x_m / fma(Float64(x_m * x_m), Float64(x_m * x_m), -1.0)) * fma(x_m, x_m, -1.0));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 84.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x \cdot x + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x + 1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x} + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2}} + 1} \]

      5. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{{x}^{2} \cdot {x}^{2} - 1 \cdot 1}{{x}^{2} - 1}}} \]

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

      16. pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 72.5

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

   6. Applied rewrites 72.5%

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

   if 2e7 < x

   1. Initial program 57.6%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 100.0

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 100.0

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;x\_m \cdot \frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 20000000.0) (* x_m (/ 1.0 (fma x_m x_m 1.0))) (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = x_m * (1.0 / fma(x_m, x_m, 1.0));
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(x_m * Float64(1.0 / fma(x_m, x_m, 1.0)));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(x$95$m * N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 84.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x + 1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x} + 1} \]

      3. lower-fma.f64 84.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \]

   4. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x + 1}} \]

      3. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(x \cdot x + 1\right)\right)}} \]

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-neg-in N/A

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

      11. pow2 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. pow2 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. pow2 N/A

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

      14. metadata-eval N/A

          \[\leadsto x \cdot \frac{1}{x \cdot x + \color{blue}{1}} \]

      15. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{1}{x \cdot x + 1}} \]

      16. lower-fma.f64 84.0

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

   8. Applied rewrites 84.0%

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

   if 2e7 < x

   1. Initial program 57.6%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 100.0

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 100.0

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 20000000.0) (/ x_m (fma x_m x_m 1.0)) (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = x_m / fma(x_m, x_m, 1.0);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(x_m / fma(x_m, x_m, 1.0));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(x$95$m / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 84.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x + 1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot x} + 1} \]

      3. lower-fma.f64 84.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \]

   4. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}} \]

   if 2e7 < x

   1. Initial program 57.6%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 100.0

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 100.0

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, x\_m \cdot x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.85) (fma (- x_m) (* x_m x_m) x_m) (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.85) {
		tmp = fma(-x_m, (x_m * x_m), x_m);
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.85)
		tmp = fma(Float64(-x_m), Float64(x_m * x_m), x_m);
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.85], N[((-x$95$m) * N[(x$95$m * x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.849999999999999978

   1. Initial program 83.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. pow2 N/A

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

      5. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \left({x}^{2} - 1\right) + x \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto {x}^{\left(\frac{6}{2}\right)} \cdot \left({x}^{2} - 1\right) + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{\left(\frac{6}{2}\right)} \cdot \left({x}^{2} - 1\right) + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{6}{2}\right)}, \color{blue}{{x}^{2} - 1}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, {x}^{\color{blue}{2}} - 1, x\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2}} - 1, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, {x}^{2} - 1 \cdot \color{blue}{1}, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}, x\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, x \cdot x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot 1, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, x \cdot x + -1 \cdot 1, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, x \cdot x + -1, x\right) \]

      16. lower-fma.f64 61.9

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(x, \color{blue}{x}, -1\right), x\right) \]

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(x, x, -1\right), x\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{x}, x, -1\right), x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, x \cdot x + \color{blue}{-1}, x\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(x \cdot x + -1\right) + \color{blue}{x} \]

      4. cube-mult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x + -1\right) + x \]

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 61.9

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

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), x \cdot x, x\right) \]

      2. lift-neg.f64 61.1

         \[\leadsto \mathsf{fma}\left(-x, x \cdot x, x\right) \]

   10. Applied rewrites 61.1%

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

   if 0.849999999999999978 < x

   1. Initial program 58.3%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 99.9

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 99.9

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) x_m (/ 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], x$95$m, N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{x} \]

   if 1 < x

   1. Initial program 58.3%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   4. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. lower-pow.f64 99.9

         \[\leadsto {x}^{\color{blue}{-1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{-1}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\color{blue}{-1}} \]

      2. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{x}} \]

      3. lower-/.f64 99.9

         \[\leadsto \frac{1}{\color{blue}{x}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.4% accurate, 20.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 78.3%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 48.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Python

def code(x):
	return 1.0 / (x + (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025051 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ x (/ 1 x))))

  (/ x (+ (* x x) 1.0)))