Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.9% → 100.0%

Time: 6.9s

Alternatives: 15

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -15000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \mathbf{elif}\;y \leq 14500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(x - \frac{\left(x - t\_0\right) - 1}{y}\right) - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x 1.0) y)))
   (if (<= y -15000.0)
     (+ (/ (fma t_0 (- (/ -1.0 y) -1.0) (- 1.0 x)) y) x)
     (if (<= y 14500.0)
       (fma (/ y (- y -1.0)) (+ -1.0 x) 1.0)
       (- x (/ (- (- x (/ (- (- x t_0) 1.0) y)) 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (x - 1.0) / y;
	double tmp;
	if (y <= -15000.0) {
		tmp = (fma(t_0, ((-1.0 / y) - -1.0), (1.0 - x)) / y) + x;
	} else if (y <= 14500.0) {
		tmp = fma((y / (y - -1.0)), (-1.0 + x), 1.0);
	} else {
		tmp = x - (((x - (((x - t_0) - 1.0) / y)) - 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (y <= -15000.0)
		tmp = Float64(Float64(fma(t_0, Float64(Float64(-1.0 / y) - -1.0), Float64(1.0 - x)) / y) + x);
	elseif (y <= 14500.0)
		tmp = fma(Float64(y / Float64(y - -1.0)), Float64(-1.0 + x), 1.0);
	else
		tmp = Float64(x - Float64(Float64(Float64(x - Float64(Float64(Float64(x - t_0) - 1.0) / y)) - 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -15000.0], N[(N[(N[(t$95$0 * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 14500.0], N[(N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(N[(x - N[(N[(N[(x - t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -15000

   1. Initial program 28.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 99.9%

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

   if -15000 < y < 14500

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.9%

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

   if 14500 < y

   1. Initial program 35.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 61.3%

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

   9. Taylor expanded in y around -inf

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

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{x - \frac{\left(x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\right) - 1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x - 1}{y}, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \mathbf{elif}\;y \leq 14500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{y}^{-1}\\ \mathbf{elif}\;t\_0 \leq 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (<= t_0 -5e-12)
     (- 1.0 (- 1.0 x))
     (if (<= t_0 2e-5)
       (pow y -1.0)
       (if (<= t_0 1e+60) (fma (- x 1.0) y 1.0) (- 1.0 (- x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= -5e-12) {
		tmp = 1.0 - (1.0 - x);
	} else if (t_0 <= 2e-5) {
		tmp = pow(y, -1.0);
	} else if (t_0 <= 1e+60) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = 1.0 - -x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)))
	tmp = 0.0
	if (t_0 <= -5e-12)
		tmp = Float64(1.0 - Float64(1.0 - x));
	elseif (t_0 <= 2e-5)
		tmp = y ^ -1.0;
	elseif (t_0 <= 1e+60)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = Float64(1.0 - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-12], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[Power[y, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 1e+60], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(1.0 - (-x)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -4.9999999999999997e-12

   1. Initial program 61.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -4.9999999999999997e-12 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2.00000000000000016e-5

   1. Initial program 9.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.1%

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

   if 2.00000000000000016e-5 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 9.9999999999999995e59

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 9.9999999999999995e59 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 51.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.5%

      \[\leadsto 1 - \left(-x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{elif}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{y}^{-1}\\ \mathbf{elif}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15000 \lor \neg \left(y \leq 14500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x - 1}{y}, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -15000.0) (not (<= y 14500.0)))
   (+ (/ (fma (/ (- x 1.0) y) (- (/ -1.0 y) -1.0) (- 1.0 x)) y) x)
   (fma (/ y (- y -1.0)) (+ -1.0 x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -15000.0) || !(y <= 14500.0)) {
		tmp = (fma(((x - 1.0) / y), ((-1.0 / y) - -1.0), (1.0 - x)) / y) + x;
	} else {
		tmp = fma((y / (y - -1.0)), (-1.0 + x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -15000.0) || !(y <= 14500.0))
		tmp = Float64(Float64(fma(Float64(Float64(x - 1.0) / y), Float64(Float64(-1.0 / y) - -1.0), Float64(1.0 - x)) / y) + x);
	else
		tmp = fma(Float64(y / Float64(y - -1.0)), Float64(-1.0 + x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -15000.0], N[Not[LessEqual[y, 14500.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15000 or 14500 < y

   1. Initial program 32.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 99.9%

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

   if -15000 < y < 14500

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15000 \lor \neg \left(y \leq 14500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x - 1}{y}, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 235000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot x - y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) - \frac{1 - x}{y}}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -130000000.0)
   (- x (/ (- x 1.0) y))
   (if (<= y 235000.0)
     (fma (/ (- (* y x) y) (fma y y -1.0)) (- y 1.0) 1.0)
     (+ (/ (- (- 1.0 x) (/ (- 1.0 x) y)) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -130000000.0) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 235000.0) {
		tmp = fma((((y * x) - y) / fma(y, y, -1.0)), (y - 1.0), 1.0);
	} else {
		tmp = (((1.0 - x) - ((1.0 - x) / y)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -130000000.0)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 235000.0)
		tmp = fma(Float64(Float64(Float64(y * x) - y) / fma(y, y, -1.0)), Float64(y - 1.0), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) - Float64(Float64(1.0 - x) / y)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -130000000.0], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 235000.0], N[(N[(N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y - 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e8

   1. Initial program 28.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.3e8 < y < 235000

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 235000 < y

   1. Initial program 34.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) + \frac{-1 + x}{y}}{y} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 235000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot x - y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) - \frac{1 - x}{y}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2300000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 480000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1 + x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) - \frac{1 - x}{y}}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2300000000000.0)
   (- x (/ (- x 1.0) y))
   (if (<= y 480000.0)
     (fma y (/ (+ -1.0 x) (- y -1.0)) 1.0)
     (+ (/ (- (- 1.0 x) (/ (- 1.0 x) y)) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2300000000000.0) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 480000.0) {
		tmp = fma(y, ((-1.0 + x) / (y - -1.0)), 1.0);
	} else {
		tmp = (((1.0 - x) - ((1.0 - x) / y)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2300000000000.0)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 480000.0)
		tmp = fma(y, Float64(Float64(-1.0 + x) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) - Float64(Float64(1.0 - x) / y)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2300000000000.0], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 480000.0], N[(y * N[(N[(-1.0 + x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e12

   1. Initial program 27.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.3e12 < y < 4.8e5

   1. Initial program 99.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 99.3%

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

   if 4.8e5 < y

   1. Initial program 34.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) + \frac{-1 + x}{y}}{y} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2300000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 480000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1 + x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) - \frac{1 - x}{y}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -125000000 \lor \neg \left(y \leq 180000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x \cdot y}{\left(-y\right) - 1} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -125000000.0) (not (<= y 180000000.0)))
   (- x (/ (- x 1.0) y))
   (+ (/ (- y (* x y)) (- (- y) 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -125000000.0) || !(y <= 180000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = ((y - (x * y)) / (-y - 1.0)) + 1.0;
	}
	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, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-125000000.0d0)) .or. (.not. (y <= 180000000.0d0))) then
        tmp = x - ((x - 1.0d0) / y)
    else
        tmp = ((y - (x * y)) / (-y - 1.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -125000000.0) || !(y <= 180000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = ((y - (x * y)) / (-y - 1.0)) + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -125000000.0) or not (y <= 180000000.0):
		tmp = x - ((x - 1.0) / y)
	else:
		tmp = ((y - (x * y)) / (-y - 1.0)) + 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -125000000.0) || !(y <= 180000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = Float64(Float64(Float64(y - Float64(x * y)) / Float64(Float64(-y) - 1.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -125000000.0) || ~((y <= 180000000.0)))
		tmp = x - ((x - 1.0) / y);
	else
		tmp = ((y - (x * y)) / (-y - 1.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -125000000.0], N[Not[LessEqual[y, 180000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision] / N[((-y) - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e8 or 1.8e8 < y

   1. Initial program 30.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.25e8 < y < 1.8e8

   1. Initial program 99.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 99.3%

       \[\leadsto \left(1 + \left(-\frac{y}{y - -1}\right)\right) + \color{blue}{\frac{y}{y - -1} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.3%

       \[\leadsto \frac{y - x \cdot y}{-\left(y - -1\right)} + \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000 \lor \neg \left(y \leq 180000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x \cdot y}{\left(-y\right) - 1} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2300000000000 \lor \neg \left(y \leq 165000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1 + x}{y - -1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2300000000000.0) (not (<= y 165000000.0)))
   (- x (/ (- x 1.0) y))
   (fma y (/ (+ -1.0 x) (- y -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2300000000000.0) || !(y <= 165000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(y, ((-1.0 + x) / (y - -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2300000000000.0) || !(y <= 165000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(y, Float64(Float64(-1.0 + x) / Float64(y - -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2300000000000.0], N[Not[LessEqual[y, 165000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(-1.0 + x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e12 or 1.65e8 < y

   1. Initial program 29.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -2.3e12 < y < 1.65e8

   1. Initial program 99.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 99.3%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1 + x}{y - -1}}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2300000000000 \lor \neg \left(y \leq 165000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1 + x}{y - -1}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 18000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{1 + y}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -210.0) (not (<= y 18000.0)))
   (- x (/ (- x 1.0) y))
   (fma y (/ x (+ 1.0 y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -210.0) || !(y <= 18000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(y, (x / (1.0 + y)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -210.0) || !(y <= 18000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(y, Float64(x / Float64(1.0 + y)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -210.0], N[Not[LessEqual[y, 18000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -210 or 18000 < y

   1. Initial program 32.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -210 < y < 18000

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.4%

       \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{1 + y}}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 18000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{1 + y}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- x (/ (- x 1.0) y))
   (fma (* y (- 1.0 x)) (+ -1.0 y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma((y * (1.0 - x)), (-1.0 + y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(Float64(y * Float64(1.0 - x)), Float64(-1.0 + y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-*r* N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. mul-1-neg N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.85)))
   (- x (/ -1.0 y))
   (fma (* y (- 1.0 x)) (+ -1.0 y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.85)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = fma((y * (1.0 - x)), (-1.0 + y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.85))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = fma(Float64(y * Float64(1.0 - x)), Float64(-1.0 + y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.85]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.849999999999999978 < y

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

      \[\leadsto x - \frac{-1}{y} \]

   if -1 < y < 0.849999999999999978

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-*r* N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. mul-1-neg N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(1 - x\right), -1 + y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.8))) (- x (/ -1.0 y)) (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.8)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.8))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.8]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

      \[\leadsto x - \frac{-1}{y} \]

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.06\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.06))) (- x (/ x y)) (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.06)) {
		tmp = x - (x / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.06))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.06]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.0600000000000001 < y

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.7%

      \[\leadsto x - \frac{x}{\color{blue}{y}} \]

   if -1 < y < 1.0600000000000001

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.06\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- 1.0 (- 1.0 x))
   (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(1.0 - x));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ 1 - \left(-x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (- x)))

C

double code(double x, double y) {
	return 1.0 - -x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y) {
	return 1.0 - -x;
}

Python

def code(x, y):
	return 1.0 - -x

Julia

function code(x, y)
	return Float64(1.0 - Float64(-x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - -x;
end

Wolfram

code[x_, y_] := N[(1.0 - (-x)), $MachinePrecision]

Derivation

1. Initial program 64.1%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. lower--.f64 32.3

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

5. Applied rewrites 32.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 56.7%

   \[\leadsto 1 - \left(-x\right) \]
9. Add Preprocessing

Alternative 15: 3.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 1.0))

C

double code(double x, double y) {
	return 1.0 - 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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 - 1.0;
}

Python

def code(x, y):
	return 1.0 - 1.0

Julia

function code(x, y)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 64.1%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. lower--.f64 32.3

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

5. Applied rewrites 32.3%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - 1 \]
7. Step-by-step derivation

8. Applied rewrites 3.1%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	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, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024352 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))