x / (x^2 + 1)

Percentage Accurate: 76.0% → 100.0%

Time: 3.1s

Alternatives: 5

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: 76.0% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(x\_m, x\_m, -1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot x\_m, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

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 100000000.0)
    (/ (* x_m (fma x_m x_m -1.0)) (fma (* x_m x_m) (* 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 <= 100000000.0) {
		tmp = (x_m * fma(x_m, x_m, -1.0)) / fma((x_m * x_m), (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 <= 100000000.0)
		tmp = Float64(Float64(x_m * fma(x_m, x_m, -1.0)) / fma(Float64(x_m * x_m), Float64(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, 100000000.0], N[(N[(x$95$m * N[(x$95$m * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 87.1%

      \[\frac{x}{x \cdot x + 1} \]
   2. 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. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-sqr-1 N/A

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

      10. difference-of-sqr--1-rev N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. pow-prod-down N/A

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

      19. pow-prod-up N/A

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

      20. lower-pow.f64 N/A

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

      21. metadata-eval 78.7

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

   3. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. difference-of-sqr--1-rev N/A

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

      10. lower-fma.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 1e8 < x

   1. Initial program 44.2%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

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 100000000.0) (/ 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 <= 100000000.0) {
		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 <= 100000000.0)
		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, 100000000.0], 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 < 1e8

   1. Initial program 87.1%

      \[\frac{x}{x \cdot x + 1} \]
   2. 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 87.1

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

   3. Applied rewrites 87.1%

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

   if 1e8 < x

   1. Initial program 44.2%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

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 0.86) (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 <= 0.86) {
		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 <= 0.86)
		tmp = fma(Float64(Float64(-x_m) * 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, 0.86], N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 86.9%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 66.2

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

   4. Applied rewrites 66.2%

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

   6. Applied rewrites 66.2%

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

   if 0.859999999999999987 < x

   1. Initial program 46.0%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

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) (/ 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 / 1.0;
	} 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 / 1.0d0
    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 / 1.0;
	} 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 / 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 <= 1.0)
		tmp = Float64(x_m / 1.0);
	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 / 1.0;
	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], N[(x$95$m / 1.0), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.9%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 66.5%

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

   if 1 < x

   1. Initial program 46.0%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 5: 52.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / 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 * (1.0d0 / 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 * (1.0 / x_m);
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / 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 * (1.0 / 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 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

   \[\frac{x}{x \cdot x + 1} \]

2. Taylor expanded in x around inf

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

   1. lower-/.f64 50.7

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

4. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
5. 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 2024363 -o localize:costs -o setup:simplify -o generate:simplify
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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