ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 88.4% → 99.5%

Time: 3.3s

Alternatives: 10

Speedup: 0.3×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := \left(x \cdot x\right) \cdot x\\ t_3 := \sqrt{t\_2 \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t\_0 - \left(t\_2 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, -t\_3, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (pow (+ x eps) 5.0))
       (t_1 (- t_0 (pow x 5.0)))
       (t_2 (* (* x x) x))
       (t_3 (sqrt (* t_2 (* x x)))))
  (if (<= t_1 -1e-320)
    (- t_0 (* (* t_2 x) x))
    (if (<= t_1 0.0)
      (* eps (* 5.0 (pow x 4.0)))
      (fma t_3 (- t_3) t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = (x * x) * x;
	double t_3 = sqrt((t_2 * (x * x)));
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_0 - ((t_2 * x) * x);
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = fma(t_3, -t_3, t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(Float64(x * x) * x)
	t_3 = sqrt(Float64(t_2 * Float64(x * x)))
	tmp = 0.0
	if (t_1 <= -1e-320)
		tmp = Float64(t_0 - Float64(Float64(t_2 * x) * x));
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = fma(t_3, Float64(-t_3), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-320], N[(t$95$0 - N[(N[(t$95$2 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * (-t$95$3) + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      2. lift-+.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      3. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      4. sum-square-pow N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\left(\left({x}^{2} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right)} - {x}^{5} \]

      5. pow2 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{x \cdot x} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      6. *-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(x \cdot x + 2 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{\left(2 \cdot \left(\varepsilon \cdot x\right) + x \cdot x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      8. associate-*r* N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{\left(2 \cdot \varepsilon\right) \cdot x} + x \cdot x\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      9. distribute-rgt-out N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{x \cdot \left(2 \cdot \varepsilon + x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      10. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, 2 \cdot \varepsilon + x, {\varepsilon}^{2}\right)} - {x}^{5} \]

      11. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, \varepsilon, x\right)}, {\varepsilon}^{2}\right) - {x}^{5} \]

      12. unpow2 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

      13. lower-*.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

   5. Applied rewrites 86.0%

      \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \varepsilon \cdot \varepsilon\right)} - {x}^{5} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.5%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2} - {x}^{5}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2} + \left(\mathsf{neg}\left({x}^{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3}} \cdot {\left(\varepsilon + x\right)}^{2} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      6. pow-prod-up N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{\left(3 + 2\right)}} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      7. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{\left(3 + 2\right)} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{\left(3 + 2\right)} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{5}} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} + \left(\mathsf{neg}\left({x}^{5}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({x}^{5}\right)\right) + {\left(x + \varepsilon\right)}^{5}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{x}^{5}}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({x}^{\color{blue}{\left(2 \cdot \frac{5}{2}\right)}}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({x}^{\left(2 \cdot \color{blue}{\frac{5}{2}}\right)}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left({x}^{\left(2 \cdot \color{blue}{\frac{5}{2}}\right)}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      17. pow-sqr N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{x}^{\left(\frac{5}{2}\right)} \cdot {x}^{\left(\frac{5}{2}\right)}}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      18. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{x}^{\left(\frac{5}{2}\right)}} \cdot {x}^{\left(\frac{5}{2}\right)}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      19. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left({x}^{\left(\frac{5}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{5}{2}\right)}}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

      20. sqr-abs-rev N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left|{x}^{\left(\frac{5}{2}\right)}\right| \cdot \left|{x}^{\left(\frac{5}{2}\right)}\right|}\right)\right) + {\left(x + \varepsilon\right)}^{5} \]

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

          \[\leadsto \color{blue}{\left|{x}^{\left(\frac{5}{2}\right)}\right| \cdot \left(\mathsf{neg}\left(\left|{x}^{\left(\frac{5}{2}\right)}\right|\right)\right)} + {\left(x + \varepsilon\right)}^{5} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}, -\sqrt{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}, {\left(x + \varepsilon\right)}^{5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t\_0 - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-x\right) \cdot x\right) \cdot x, x \cdot x, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
  (if (<= t_1 -1e-320)
    (- t_0 (* (* (* (* x x) x) x) x))
    (if (<= t_1 0.0)
      (* eps (* 5.0 (pow x 4.0)))
      (fma (* (* (- x) x) x) (* x x) t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_0 - ((((x * x) * x) * x) * x);
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = fma(((-x * x) * x), (x * x), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-320)
		tmp = Float64(t_0 - Float64(Float64(Float64(Float64(x * x) * x) * x) * x));
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = fma(Float64(Float64(Float64(-x) * x) * x), Float64(x * x), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-320], N[(t$95$0 - N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      2. lift-+.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      3. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      4. sum-square-pow N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\left(\left({x}^{2} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right)} - {x}^{5} \]

      5. pow2 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{x \cdot x} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      6. *-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(x \cdot x + 2 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{\left(2 \cdot \left(\varepsilon \cdot x\right) + x \cdot x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      8. associate-*r* N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{\left(2 \cdot \varepsilon\right) \cdot x} + x \cdot x\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      9. distribute-rgt-out N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{x \cdot \left(2 \cdot \varepsilon + x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      10. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, 2 \cdot \varepsilon + x, {\varepsilon}^{2}\right)} - {x}^{5} \]

      11. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, \varepsilon, x\right)}, {\varepsilon}^{2}\right) - {x}^{5} \]

      12. unpow2 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

      13. lower-*.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

   5. Applied rewrites 86.0%

      \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \varepsilon \cdot \varepsilon\right)} - {x}^{5} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.5%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      2. lift-+.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      3. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      4. sum-square-pow N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\left(\left({x}^{2} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right)} - {x}^{5} \]

      5. pow2 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{x \cdot x} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      6. *-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(x \cdot x + 2 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{\left(2 \cdot \left(\varepsilon \cdot x\right) + x \cdot x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      8. associate-*r* N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{\left(2 \cdot \varepsilon\right) \cdot x} + x \cdot x\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      9. distribute-rgt-out N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{x \cdot \left(2 \cdot \varepsilon + x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      10. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, 2 \cdot \varepsilon + x, {\varepsilon}^{2}\right)} - {x}^{5} \]

      11. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, \varepsilon, x\right)}, {\varepsilon}^{2}\right) - {x}^{5} \]

      12. unpow2 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

      13. lower-*.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

   5. Applied rewrites 86.0%

      \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \varepsilon \cdot \varepsilon\right)} - {x}^{5} \]

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-x\right) \cdot x\right) \cdot x, x \cdot x, {\left(x + \varepsilon\right)}^{5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (pow (+ x eps) 5.0))
       (t_1 (- t_0 (pow x 5.0)))
       (t_2 (- t_0 (* (* (* (* x x) x) x) x))))
  (if (<= t_1 -1e-320)
    t_2
    (if (<= t_1 0.0) (* eps (* 5.0 (pow x 4.0))) t_2))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = t_0 - ((((x * x) * x) * x) * x);
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((((x * x) * x) * x) * x)
    if (t_1 <= (-1d-320)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 - ((((x * x) * x) * x) * x);
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((((x * x) * x) * x) * x)
	tmp = 0
	if t_1 <= -1e-320:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(Float64(Float64(x * x) * x) * x) * x))
	tmp = 0.0
	if (t_1 <= -1e-320)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((((x * x) * x) * x) * x);
	tmp = 0.0;
	if (t_1 <= -1e-320)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-320], t$95$2, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      2. lift-+.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      3. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      4. sum-square-pow N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\left(\left({x}^{2} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right)} - {x}^{5} \]

      5. pow2 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{x \cdot x} + 2 \cdot \left(x \cdot \varepsilon\right)\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      6. *-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(x \cdot x + 2 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{\left(2 \cdot \left(\varepsilon \cdot x\right) + x \cdot x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      8. associate-*r* N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\left(\color{blue}{\left(2 \cdot \varepsilon\right) \cdot x} + x \cdot x\right) + {\varepsilon}^{2}\right) - {x}^{5} \]

      9. distribute-rgt-out N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \left(\color{blue}{x \cdot \left(2 \cdot \varepsilon + x\right)} + {\varepsilon}^{2}\right) - {x}^{5} \]

      10. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, 2 \cdot \varepsilon + x, {\varepsilon}^{2}\right)} - {x}^{5} \]

      11. lower-fma.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, \varepsilon, x\right)}, {\varepsilon}^{2}\right) - {x}^{5} \]

      12. unpow2 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

      13. lower-*.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \color{blue}{\varepsilon \cdot \varepsilon}\right) - {x}^{5} \]

   5. Applied rewrites 86.0%

      \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, \varepsilon, x\right), \varepsilon \cdot \varepsilon\right)} - {x}^{5} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.5%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (pow (+ x eps) 5.0))
       (t_1 (- t_0 (pow x 5.0)))
       (t_2 (- t_0 (* (* (* x x) x) (* x x)))))
  (if (<= t_1 -1e-320)
    t_2
    (if (<= t_1 0.0) (* eps (* 5.0 (pow x 4.0))) t_2))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = t_0 - (((x * x) * x) * (x * x));
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - (((x * x) * x) * (x * x))
    if (t_1 <= (-1d-320)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 - (((x * x) * x) * (x * x));
	double tmp;
	if (t_1 <= -1e-320) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - (((x * x) * x) * (x * x))
	tmp = 0
	if t_1 <= -1e-320:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(Float64(x * x) * x) * Float64(x * x)))
	tmp = 0.0
	if (t_1 <= -1e-320)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - (((x * x) * x) * (x * x));
	tmp = 0.0;
	if (t_1 <= -1e-320)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-320], t$95$2, If[LessEqual[t$95$1, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3} \cdot {\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

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

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{3}} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      7. +-commutative N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      8. lower-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{3} \cdot {\left(x + \varepsilon\right)}^{2} - {x}^{5} \]

      9. lower-unsound-pow.f64 86.0%

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(x + \varepsilon\right)}^{2}} - {x}^{5} \]

      10. lift-+.f64 N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(x + \varepsilon\right)}}^{2} - {x}^{5} \]

      11. +-commutative N/A

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

      12. lower-+.f64 86.0%

          \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot {\color{blue}{\left(\varepsilon + x\right)}}^{2} - {x}^{5} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3} \cdot {\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{3}} \cdot {\left(\varepsilon + x\right)}^{2} - {x}^{5} \]

      3. lift-pow.f64 N/A

         \[\leadsto {\left(\varepsilon + x\right)}^{3} \cdot \color{blue}{{\left(\varepsilon + x\right)}^{2}} - {x}^{5} \]

      4. pow-prod-up N/A

         \[\leadsto \color{blue}{{\left(\varepsilon + x\right)}^{\left(3 + 2\right)}} - {x}^{5} \]

      5. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\varepsilon + x\right)}}^{\left(3 + 2\right)} - {x}^{5} \]

      6. +-commutative N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{\left(3 + 2\right)} - {x}^{5} \]

      7. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{5}} - {x}^{5} \]

      8. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      9. lift-+.f64 88.4%

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      10. lift-pow.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{5}} \]

      11. metadata-eval N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      12. pow-add N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      13. lower-unsound-pow.f32 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      14. lower-pow.f32 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      15. cube-unmult N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{2} \]

      16. lower-unsound-pow.f32 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      17. lower-pow.f32 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      18. pow2 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

      20. cube-unmult N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3}} \cdot \left(x \cdot x\right) \]

      21. pow3 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      22. lower-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      23. lower-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot \left(x \cdot x\right) \]

      24. lower-*.f64 84.9%

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;{\varepsilon}^{5} - \left|t\_1\right|\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} - t\_1\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
       (t_1 (* (* (* x x) x) (* x x))))
  (if (<= t_0 -1e-320)
    (- (pow eps 5.0) (fabs t_1))
    (if (<= t_0 0.0)
      (* eps (* 5.0 (pow x 4.0)))
      (- (pow eps 5.0) t_1)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = ((x * x) * x) * (x * x);
	double tmp;
	if (t_0 <= -1e-320) {
		tmp = pow(eps, 5.0) - fabs(t_1);
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0) - t_1;
	}
	return tmp;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((x * x) * x) * (x * x)
    if (t_0 <= (-1d-320)) then
        tmp = (eps ** 5.0d0) - abs(t_1)
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = (eps ** 5.0d0) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = ((x * x) * x) * (x * x);
	double tmp;
	if (t_0 <= -1e-320) {
		tmp = Math.pow(eps, 5.0) - Math.abs(t_1);
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0) - t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = ((x * x) * x) * (x * x)
	tmp = 0
	if t_0 <= -1e-320:
		tmp = math.pow(eps, 5.0) - math.fabs(t_1)
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0) - t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(x * x) * x) * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e-320)
		tmp = Float64((eps ^ 5.0) - abs(t_1));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = Float64((eps ^ 5.0) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = ((x * x) * x) * (x * x);
	tmp = 0.0;
	if (t_0 <= -1e-320)
		tmp = (eps ^ 5.0) - abs(t_1);
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = (eps ^ 5.0) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-320], N[(N[Power[eps, 5.0], $MachinePrecision] - N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   3. Step-by-step derivation

   4. Applied rewrites 79.7%

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\sqrt{{x}^{5}} \cdot \sqrt{{x}^{5}}} \]

      2. sqrt-unprod N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\sqrt{{x}^{5} \cdot {x}^{5}}} \]

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

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\left|{x}^{5}\right|} \]

      4. lower-fabs.f64 79.7%

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\left|{x}^{5}\right|} \]

      5. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{{x}^{5}}\right| \]

      6. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} - \left|{x}^{\color{blue}{\left(3 + 2\right)}}\right| \]

      7. pow-add N/A

         \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{{x}^{3} \cdot {x}^{2}}\right| \]

      8. lower-unsound-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{{x}^{3}} \cdot {x}^{2}\right| \]

      9. lower-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{{x}^{3}} \cdot {x}^{2}\right| \]

      10. cube-unmult N/A

          \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{2}\right| \]

      11. lower-unsound-pow.f32 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}}\right| \]

      12. lower-pow.f32 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}}\right| \]

      13. pow2 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right| \]

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

          \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right| \]

      15. cube-unmult N/A

          \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right| \]

      16. pow3 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right)\right| \]

      17. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right)\right| \]

      18. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \left|\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot \left(x \cdot x\right)\right| \]

      19. lower-*.f64 79.7%

          \[\leadsto {\varepsilon}^{5} - \left|\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right| \]

   6. Applied rewrites 79.7%

      \[\leadsto {\varepsilon}^{5} - \color{blue}{\left|\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right|} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   3. Step-by-step derivation

   4. Applied rewrites 79.7%

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{5}} \]

      2. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      3. pow-add N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      4. lower-unsound-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      5. lower-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      6. cube-unmult N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{2} \]

      7. lower-unsound-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      8. lower-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      9. pow2 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

      11. cube-unmult N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot \left(x \cdot x\right) \]

      12. pow3 N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot \left(x \cdot x\right) \]

      15. lower-*.f64 79.7%

          \[\leadsto {\varepsilon}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Applied rewrites 79.7%

      \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := {\varepsilon}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
       (t_1 (- (pow eps 5.0) (* (* (* x x) x) (* x x)))))
  (if (<= t_0 -1e-320)
    t_1
    (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = pow(eps, 5.0) - (((x * x) * x) * (x * x));
	double tmp;
	if (t_0 <= -1e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps ** 5.0d0) - (((x * x) * x) * (x * x))
    if (t_0 <= (-1d-320)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = Math.pow(eps, 5.0) - (((x * x) * x) * (x * x));
	double tmp;
	if (t_0 <= -1e-320) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = math.pow(eps, 5.0) - (((x * x) * x) * (x * x))
	tmp = 0
	if t_0 <= -1e-320:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64((eps ^ 5.0) - Float64(Float64(Float64(x * x) * x) * Float64(x * x)))
	tmp = 0.0
	if (t_0 <= -1e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps ^ 5.0) - (((x * x) * x) * (x * x));
	tmp = 0.0;
	if (t_0 <= -1e-320)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[eps, 5.0], $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-320], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9998886718268301e-321 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   3. Step-by-step derivation

   4. Applied rewrites 79.7%

      \[\leadsto {\color{blue}{\varepsilon}}^{5} - {x}^{5} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{5}} \]

      2. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      3. pow-add N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      4. lower-unsound-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      5. lower-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot {x}^{2} \]

      6. cube-unmult N/A

         \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{2} \]

      7. lower-unsound-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      8. lower-pow.f32 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{{x}^{2}} \]

      9. pow2 N/A

         \[\leadsto {\varepsilon}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

      11. cube-unmult N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{{x}^{3}} \cdot \left(x \cdot x\right) \]

      12. pow3 N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{5} - \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot \left(x \cdot x\right) \]

      15. lower-*.f64 79.7%

          \[\leadsto {\varepsilon}^{5} - \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Applied rewrites 79.7%

      \[\leadsto {\varepsilon}^{5} - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \]

   if -9.9998886718268301e-321 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

      2. lower-pow.f64 82.5%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   7. Applied rewrites 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.5% accurate, 1.7× speedup

Math

\[\varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* eps (* 5.0 (pow x 4.0))))

C

double code(double x, double eps) {
	return eps * (5.0 * pow(x, 4.0));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (5.0d0 * (x ** 4.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (5.0 * Math.pow(x, 4.0));
}

Python

def code(x, eps):
	return eps * (5.0 * math.pow(x, 4.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(5.0 * (x ^ 4.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (5.0 * (x ^ 4.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

4. Applied rewrites 82.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

   2. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

7. Applied rewrites 82.5%

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
8. Add Preprocessing

Alternative 8: 82.5% accurate, 2.5× speedup

Math

\[x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(5 \cdot \varepsilon\right) \cdot x\right)\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* x (* (* x x) (* (* 5.0 eps) x))))

C

double code(double x, double eps) {
	return x * ((x * x) * ((5.0 * eps) * 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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * ((x * x) * ((5.0d0 * eps) * x))
end function

Java

public static double code(double x, double eps) {
	return x * ((x * x) * ((5.0 * eps) * x));
}

Python

def code(x, eps):
	return x * ((x * x) * ((5.0 * eps) * x))

Julia

function code(x, eps)
	return Float64(x * Float64(Float64(x * x) * Float64(Float64(5.0 * eps) * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = x * ((x * x) * ((5.0 * eps) * x));
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 82.5%

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

4. Applied rewrites 82.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

      \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   3. pow-plus-rev N/A

      \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{\varepsilon} + 4 \cdot \varepsilon\right) \]

   4. lower-unsound-pow.f32 N/A

      \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   5. lower-pow.f32 N/A

      \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   6. cube-unmult N/A

      \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

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

      \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\varepsilon} + 4 \cdot \varepsilon\right) \]

   8. cube-unmult N/A

      \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   9. pow3 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   11. lower-*.f64 82.5%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \varepsilon\right) \]

   12. lift-+.f64 N/A

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + \color{blue}{4 \cdot \varepsilon}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \color{blue}{\varepsilon}\right) \]

   14. distribute-rgt1-in N/A

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

   15. metadata-eval N/A

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

   16. lower-*.f64 82.5%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

6. Applied rewrites 82.5%

   \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right) \]

   5. distribute-rgt1-in N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + \color{blue}{4 \cdot \varepsilon}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + 4 \cdot \color{blue}{\varepsilon}\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\varepsilon + \color{blue}{4 \cdot \varepsilon}\right) \]

   8. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{x} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{x}\right)\right) \]

   15. lower-*.f64 82.5%

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{x}\right)\right) \]

   16. lift-+.f64 N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot x\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot x\right)\right) \]

   18. distribute-rgt1-in N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot x\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(5 \cdot \varepsilon\right) \cdot x\right)\right) \]

   20. lift-*.f64 82.5%

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(5 \cdot \varepsilon\right) \cdot x\right)\right) \]

8. Applied rewrites 82.5%

   \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(5 \cdot \varepsilon\right) \cdot x\right)\right)} \]
9. Add Preprocessing

Alternative 9: 82.5% accurate, 2.5× speedup

Math

\[\varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot 5\right)\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* eps (* (* (* x x) x) (* x 5.0))))

C

double code(double x, double eps) {
	return eps * (((x * x) * x) * (x * 5.0));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((x * x) * x) * (x * 5.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (((x * x) * x) * (x * 5.0));
}

Python

def code(x, eps):
	return eps * (((x * x) * x) * (x * 5.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(x * x) * x) * Float64(x * 5.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((x * x) * x) * (x * 5.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

4. Applied rewrites 82.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

   2. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

7. Applied rewrites 82.5%

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left({x}^{4} \cdot 5\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({x}^{\left(3 + 1\right)} \cdot 5\right) \]

   5. pow-plus N/A

      \[\leadsto \varepsilon \cdot \left(\left({x}^{3} \cdot x\right) \cdot 5\right) \]

   6. pow3 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \]

   9. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{5}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{5}\right)\right) \]

   11. lower-*.f64 82.5%

       \[\leadsto \varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot 5\right)\right) \]

9. Applied rewrites 82.5%

   \[\leadsto \varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{5}\right)\right) \]
10. Add Preprocessing

Alternative 10: 82.5% accurate, 2.5× speedup

Math

\[\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* eps (* (* x x) (* (* x x) 5.0))))

C

double code(double x, double eps) {
	return eps * ((x * x) * ((x * x) * 5.0));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((x * x) * ((x * x) * 5.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * ((x * x) * ((x * x) * 5.0));
}

Python

def code(x, eps):
	return eps * ((x * x) * ((x * x) * 5.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(x * x) * Float64(Float64(x * x) * 5.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * x) * ((x * x) * 5.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, {x}^{4}\right) \]

4. Applied rewrites 82.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

   2. lower-pow.f64 82.5%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

7. Applied rewrites 82.5%

   \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{4}}\right) \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left({x}^{4} \cdot 5\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot 5\right) \]

   5. pow-sqr N/A

      \[\leadsto \varepsilon \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot 5\right) \]

   6. pow-prod-down N/A

      \[\leadsto \varepsilon \cdot \left({\left(x \cdot x\right)}^{2} \cdot 5\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\left(x \cdot x\right)}^{2} \cdot 5\right) \]

   8. pow2 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \]

   9. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{5}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{5}\right)\right) \]

   11. lower-*.f64 82.5%

       \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5\right)\right) \]

9. Applied rewrites 82.5%

   \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{5}\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))