peicewise-defined

Percentage Accurate: 75.6% → 76.4%

Time: 3.1s

Alternatives: 5

Speedup: 1.5×

Specification

\[p \geq -1.79 \cdot 10^{+308} \land p \leq 1.79 \cdot 10^{+308}\]

Math

\[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]

FPCore

(FPCore (p)
  :precision binary64
  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))

C

double code(double p) {
	return (p + 1.0) - (pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = (p + 1.0d0) - ((fmin(p, 0.0d0) ** 2.0d0) / (fmin(p, 0.0d0) - 1.0d0))
end function

Java

public static double code(double p) {
	return (p + 1.0) - (Math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Python

def code(p):
	return (p + 1.0) - (math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0))

Julia

function code(p)
	return Float64(Float64(p + 1.0) - Float64((fmin(p, 0.0) ^ 2.0) / Float64(fmin(p, 0.0) - 1.0)))
end

MATLAB

function tmp = code(p)
	tmp = (p + 1.0) - ((min(p, 0.0) ^ 2.0) / (min(p, 0.0) - 1.0));
end

Wolfram

code[p_] := N[(N[(p + 1.0), $MachinePrecision] - N[(N[Power[N[Min[p, 0.0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Min[p, 0.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.6% accurate, 1.0× speedup

Math

\[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]

FPCore

(FPCore (p)
  :precision binary64
  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))

C

double code(double p) {
	return (p + 1.0) - (pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = (p + 1.0d0) - ((fmin(p, 0.0d0) ** 2.0d0) / (fmin(p, 0.0d0) - 1.0d0))
end function

Java

public static double code(double p) {
	return (p + 1.0) - (Math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Python

def code(p):
	return (p + 1.0) - (math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0))

Julia

function code(p)
	return Float64(Float64(p + 1.0) - Float64((fmin(p, 0.0) ^ 2.0) / Float64(fmin(p, 0.0) - 1.0)))
end

MATLAB

function tmp = code(p)
	tmp = (p + 1.0) - ((min(p, 0.0) ^ 2.0) / (min(p, 0.0) - 1.0));
end

Wolfram

code[p_] := N[(N[(p + 1.0), $MachinePrecision] - N[(N[Power[N[Min[p, 0.0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Min[p, 0.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.4% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right) \]

FPCore

(FPCore (p)
  :precision binary64
  (fma (fmin 0.0 p) (/ (fmin 0.0 p) (- 1.0 (fmin 0.0 p))) (- p -1.0)))

C

double code(double p) {
	return fma(fmin(0.0, p), (fmin(0.0, p) / (1.0 - fmin(0.0, p))), (p - -1.0));
}

Julia

function code(p)
	return fma(fmin(0.0, p), Float64(fmin(0.0, p) / Float64(1.0 - fmin(0.0, p))), Float64(p - -1.0))
end

Wolfram

code[p_] := N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(p - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) + \left(p + 1\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right)} \]

   10. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   11. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   12. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}}, p + 1\right) \]

   14. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   15. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   16. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   17. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)}, p + 1\right) \]

   18. sub-negate-rev N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   19. lower--.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   20. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   21. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   22. lower-fmin.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   23. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p + 1}\right) \]

   24. add-flip N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   25. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

3. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right)} \]
4. Add Preprocessing

Alternative 2: 76.2% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p\right) - -1 \]

FPCore

(FPCore (p)
  :precision binary64
  (- (fma (fmin 0.0 p) (/ (fmin 0.0 p) (- 1.0 (fmin 0.0 p))) p) -1.0))

C

double code(double p) {
	return fma(fmin(0.0, p), (fmin(0.0, p) / (1.0 - fmin(0.0, p))), p) - -1.0;
}

Julia

function code(p)
	return Float64(fma(fmin(0.0, p), Float64(fmin(0.0, p) / Float64(1.0 - fmin(0.0, p))), p) - -1.0)
end

Wolfram

code[p_] := N[(N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + p), $MachinePrecision] - -1.0), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \color{blue}{\left(p + 1\right)} \]

   5. add-flip N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \color{blue}{\left(p - \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   6. associate-+r- N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + p\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(p + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   8. sub-flip-reverse N/A

      \[\leadsto \color{blue}{\left(p - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(p - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

3. Applied rewrites 76.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p\right) - -1} \]
4. Add Preprocessing

Alternative 3: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(0, p\right) - 1\\ \mathbf{if}\;p \leq 17.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot p}{t\_0}\\ \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (let* ((t_0 (- (fmin 0.0 p) 1.0)))
  (if (<= p 17.5)
    (fma (fmin 0.0 p) (/ (fmin 0.0 p) (- 1.0 (fmin 0.0 p))) 1.0)
    (/ (* t_0 p) t_0))))

C

double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	double tmp;
	if (p <= 17.5) {
		tmp = fma(fmin(0.0, p), (fmin(0.0, p) / (1.0 - fmin(0.0, p))), 1.0);
	} else {
		tmp = (t_0 * p) / t_0;
	}
	return tmp;
}

Julia

function code(p)
	t_0 = Float64(fmin(0.0, p) - 1.0)
	tmp = 0.0
	if (p <= 17.5)
		tmp = fma(fmin(0.0, p), Float64(fmin(0.0, p) / Float64(1.0 - fmin(0.0, p))), 1.0);
	else
		tmp = Float64(Float64(t_0 * p) / t_0);
	end
	return tmp
end

Wolfram

code[p_] := Block[{t$95$0 = N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[p, 17.5], N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 * p), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if p < 17.5

   1. Initial program 75.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) + \left(p + 1\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right)} \]

      10. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      11. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      12. lower-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}}, p + 1\right) \]

      14. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      15. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      16. lower-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      17. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)}, p + 1\right) \]

      18. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      19. lower--.f64 76.4%

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      20. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      21. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

      22. lower-fmin.f64 76.4%

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

      23. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p + 1}\right) \]

      24. add-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      25. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right)} \]

   4. Taylor expanded in p around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{1}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{1}\right) \]

   if 17.5 < p

   1. Initial program 75.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) + \left(p + 1\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right)} \]

      10. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      11. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      12. lower-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}}, p + 1\right) \]

      14. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      15. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      16. lower-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

      17. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)}, p + 1\right) \]

      18. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      19. lower--.f64 76.4%

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      20. lift-fmin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

      21. fmin-swap N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

      22. lower-fmin.f64 76.4%

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

      23. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p + 1}\right) \]

      24. add-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      25. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right)} \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{\left(-1 - p\right) \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right) - \mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   6. Taylor expanded in p around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right)} - 1} \]

      4. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

      5. lower-fmin.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

      6. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - \color{blue}{1}} \]

      7. lower-fmin.f64 26.3%

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - \color{blue}{1}\right)\right)} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{p \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - \color{blue}{1}\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{p \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p\right)}{\mathsf{neg}\left(\left(\color{blue}{\mathsf{min}\left(p, 0\right)} - 1\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p\right)}{\mathsf{neg}\left(\left(\color{blue}{\mathsf{min}\left(p, 0\right)} - 1\right)\right)} \]

      12. frac-2neg-rev N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

      13. lower-/.f64 26.3%

          \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

      14. lift-fmin.f64 N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

      15. fmin-swap N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

      16. lift-fmin.f64 26.3%

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

      17. lift-fmin.f64 N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

      18. fmin-swap N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

      19. lift-fmin.f64 26.3%

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

   10. Applied rewrites 26.3%

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 26.3% accurate, 1.8× speedup

Math

\[\left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1} \]

FPCore

(FPCore (p)
  :precision binary64
  (* (- p) (/ (- 1.0 (fmin 0.0 p)) (- (fmin 0.0 p) 1.0))))

C

double code(double p) {
	return -p * ((1.0 - fmin(0.0, p)) / (fmin(0.0, p) - 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = -p * ((1.0d0 - fmin(0.0d0, p)) / (fmin(0.0d0, p) - 1.0d0))
end function

Java

public static double code(double p) {
	return -p * ((1.0 - fmin(0.0, p)) / (fmin(0.0, p) - 1.0));
}

Python

def code(p):
	return -p * ((1.0 - fmin(0.0, p)) / (fmin(0.0, p) - 1.0))

Julia

function code(p)
	return Float64(Float64(-p) * Float64(Float64(1.0 - fmin(0.0, p)) / Float64(fmin(0.0, p) - 1.0)))
end

MATLAB

function tmp = code(p)
	tmp = -p * ((1.0 - min(0.0, p)) / (min(0.0, p) - 1.0));
end

Wolfram

code[p_] := N[((-p) * N[(N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision] / N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) + \left(p + 1\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right)} \]

   10. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   11. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   12. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}}, p + 1\right) \]

   14. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   15. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   16. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   17. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)}, p + 1\right) \]

   18. sub-negate-rev N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   19. lower--.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   20. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   21. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   22. lower-fmin.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   23. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p + 1}\right) \]

   24. add-flip N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   25. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

3. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right)} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{\left(-1 - p\right) \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right) - \mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

6. Taylor expanded in p around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

   3. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right)} - 1} \]

   4. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

   5. lower-fmin.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

   6. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - \color{blue}{1}} \]

   7. lower-fmin.f64 26.3%

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

8. Applied rewrites 26.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(p \cdot \frac{1 - \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

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

      \[\leadsto \left(\mathsf{neg}\left(p\right)\right) \cdot \color{blue}{\frac{1 - \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   7. lift-neg.f64 N/A

      \[\leadsto \left(-p\right) \cdot \frac{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}{\mathsf{min}\left(p, 0\right) - 1} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(-p\right) \cdot \color{blue}{\frac{1 - \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   9. lower-/.f64 26.3%

      \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(p, 0\right)}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

   10. lift-fmin.f64 N/A

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

   11. fmin-swap N/A

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

   12. lift-fmin.f64 26.3%

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

   13. lift-fmin.f64 N/A

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

   14. fmin-swap N/A

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1} \]

   15. lift-fmin.f64 26.3%

       \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1} \]

10. Applied rewrites 26.3%

    \[\leadsto \left(-p\right) \cdot \color{blue}{\frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
11. Add Preprocessing

Alternative 5: 26.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(0, p\right) - 1\\ \frac{t\_0 \cdot p}{t\_0} \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (let* ((t_0 (- (fmin 0.0 p) 1.0))) (/ (* t_0 p) t_0)))

C

double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	return (t_0 * p) / t_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(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8) :: t_0
    t_0 = fmin(0.0d0, p) - 1.0d0
    code = (t_0 * p) / t_0
end function

Java

public static double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	return (t_0 * p) / t_0;
}

Python

def code(p):
	t_0 = fmin(0.0, p) - 1.0
	return (t_0 * p) / t_0

Julia

function code(p)
	t_0 = Float64(fmin(0.0, p) - 1.0)
	return Float64(Float64(t_0 * p) / t_0)
end

MATLAB

function tmp = code(p)
	t_0 = min(0.0, p) - 1.0;
	tmp = (t_0 * p) / t_0;
end

Wolfram

code[p_] := Block[{t$95$0 = N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]}, N[(N[(t$95$0 * p), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 75.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) + \left(p + 1\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} + \left(p + 1\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} + \left(p + 1\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right)} \]

   10. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   11. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   12. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}}, p + 1\right) \]

   14. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   15. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   16. lower-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}, p + 1\right) \]

   17. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)}, p + 1\right) \]

   18. sub-negate-rev N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   19. lower--.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   20. lift-fmin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}}, p + 1\right) \]

   21. fmin-swap N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   22. lower-fmin.f64 76.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}}, p + 1\right) \]

   23. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p + 1}\right) \]

   24. add-flip N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   25. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \color{blue}{p - \left(\mathsf{neg}\left(1\right)\right)}\right) \]

3. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, p - -1\right)} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{\left(-1 - p\right) \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right) - \mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

6. Taylor expanded in p around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

   3. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{min}\left(p, 0\right)} - 1} \]

   4. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, \color{blue}{0}\right) - 1} \]

   5. lower-fmin.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

   6. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - \color{blue}{1}} \]

   7. lower-fmin.f64 26.3%

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1} \]

8. Applied rewrites 26.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{min}\left(p, 0\right) - 1}\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{p \cdot \left(1 - \mathsf{min}\left(p, 0\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - \color{blue}{1}\right)\right)} \]

   7. sub-negate-rev N/A

      \[\leadsto \frac{p \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - \color{blue}{1}\right)\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{p \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} \]

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

      \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p\right)}{\mathsf{neg}\left(\left(\color{blue}{\mathsf{min}\left(p, 0\right)} - 1\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p\right)}{\mathsf{neg}\left(\left(\color{blue}{\mathsf{min}\left(p, 0\right)} - 1\right)\right)} \]

   12. frac-2neg-rev N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

   13. lower-/.f64 26.3%

       \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(p, 0\right) - 1}} \]

   14. lift-fmin.f64 N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(p, 0\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

   15. fmin-swap N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

   16. lift-fmin.f64 26.3%

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

   17. lift-fmin.f64 N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(p, 0\right) - 1} \]

   18. fmin-swap N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

   19. lift-fmin.f64 26.3%

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

10. Applied rewrites 26.3%

    \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]
11. Add Preprocessing

Developer Target 1: 52.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;p \geq -1 \cdot 10^{-6} \land p \leq 0:\\ \;\;\;\;\left(p + 1\right) - \frac{p \cdot p}{p - 1}\\ \mathbf{else}:\\ \;\;\;\;p - \mathsf{fma}\left(\frac{p}{p - 1}, p, -1\right)\\ \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (if (and (>= p -1e-6) (<= p 0.0))
  (- (+ p 1.0) (/ (* p p) (- p 1.0)))
  (- p (fma (/ p (- p 1.0)) p -1.0))))

C

double code(double p) {
	double tmp;
	if ((p >= -1e-6) && (p <= 0.0)) {
		tmp = (p + 1.0) - ((p * p) / (p - 1.0));
	} else {
		tmp = p - fma((p / (p - 1.0)), p, -1.0);
	}
	return tmp;
}

Julia

function code(p)
	tmp = 0.0
	if ((p >= -1e-6) && (p <= 0.0))
		tmp = Float64(Float64(p + 1.0) - Float64(Float64(p * p) / Float64(p - 1.0)));
	else
		tmp = Float64(p - fma(Float64(p / Float64(p - 1.0)), p, -1.0));
	end
	return tmp
end

Wolfram

code[p_] := If[And[GreaterEqual[p, -1e-6], LessEqual[p, 0.0]], N[(N[(p + 1.0), $MachinePrecision] - N[(N[(p * p), $MachinePrecision] / N[(p - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(p - N[(N[(p / N[(p - 1.0), $MachinePrecision]), $MachinePrecision] * p + -1.0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025210 
(FPCore (p)
  :name "peicewise-defined"
  :precision binary64
  :pre (and (>= p -1.79e+308) (<= p 1.79e+308))

  :alt
  (if (and (>= p -1/1000000) (<= p 0)) (- (+ p 1) (/ (* p p) (- p 1))) (- p (fma (/ p (- p 1)) p -1)))

  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))