Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 11

Speedup: 1.0×

• 75.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[e^{\left(x \cdot y\right) \cdot y} \]

FPCore

(FPCore (x y)
  :precision binary64
  (exp (* (* x y) y)))

C

double code(double x, double y) {
	return exp(((x * y) * y));
}

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 = exp(((x * y) * y))
end function

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = exp(((x * y) * y));
end

Wolfram

code[x_, y_] := N[Exp[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[e^{\left(x \cdot y\right) \cdot y} \]

FPCore

(FPCore (x y)
  :precision binary64
  (exp (* (* x y) y)))

C

double code(double x, double y) {
	return exp(((x * y) * y));
}

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 = exp(((x * y) * y))
end function

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = exp(((x * y) * y));
end

Wolfram

code[x_, y_] := N[Exp[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]

Alternative 1: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot y\right) \cdot y\\ t_1 := \left(y \cdot x\right) \cdot y\\ t_2 := \left(\left(y \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{-1}{t\_1 - 1}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\frac{\left(\left(t\_1 \cdot y\right) \cdot x\right) \cdot y - 1}{t\_1 - -1}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \sqrt{\sqrt{t\_2 \cdot t\_2}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* x y) y))
       (t_1 (* (* y x) y))
       (t_2 (* (* (* y y) y) y)))
  (if (<= t_0 (- INFINITY))
    (/ -1.0 (- t_1 1.0))
    (if (<= t_0 4e-23)
      (/ -1.0 (/ (- (* (* (* t_1 y) x) y) 1.0) (- t_1 -1.0)))
      (+ 1.0 (* x (sqrt (sqrt (* t_2 t_2)))))))))

C

double code(double x, double y) {
	double t_0 = (x * y) * y;
	double t_1 = (y * x) * y;
	double t_2 = ((y * y) * y) * y;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -1.0 / (t_1 - 1.0);
	} else if (t_0 <= 4e-23) {
		tmp = -1.0 / (((((t_1 * y) * x) * y) - 1.0) / (t_1 - -1.0));
	} else {
		tmp = 1.0 + (x * sqrt(sqrt((t_2 * t_2))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x * y) * y;
	double t_1 = (y * x) * y;
	double t_2 = ((y * y) * y) * y;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 / (t_1 - 1.0);
	} else if (t_0 <= 4e-23) {
		tmp = -1.0 / (((((t_1 * y) * x) * y) - 1.0) / (t_1 - -1.0));
	} else {
		tmp = 1.0 + (x * Math.sqrt(Math.sqrt((t_2 * t_2))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) * y
	t_1 = (y * x) * y
	t_2 = ((y * y) * y) * y
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -1.0 / (t_1 - 1.0)
	elif t_0 <= 4e-23:
		tmp = -1.0 / (((((t_1 * y) * x) * y) - 1.0) / (t_1 - -1.0))
	else:
		tmp = 1.0 + (x * math.sqrt(math.sqrt((t_2 * t_2))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * y)
	t_1 = Float64(Float64(y * x) * y)
	t_2 = Float64(Float64(Float64(y * y) * y) * y)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(t_1 - 1.0));
	elseif (t_0 <= 4e-23)
		tmp = Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(t_1 * y) * x) * y) - 1.0) / Float64(t_1 - -1.0)));
	else
		tmp = Float64(1.0 + Float64(x * sqrt(sqrt(Float64(t_2 * t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) * y;
	t_1 = (y * x) * y;
	t_2 = ((y * y) * y) * y;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -1.0 / (t_1 - 1.0);
	elseif (t_0 <= 4e-23)
		tmp = -1.0 / (((((t_1 * y) * x) * y) - 1.0) / (t_1 - -1.0));
	else
		tmp = 1.0 + (x * sqrt(sqrt((t_2 * t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(-1.0 / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-23], N[(-1.0 / N[(N[(N[(N[(N[(t$95$1 * y), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -inf.0

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 63.3%

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

   11. Applied rewrites 63.3%

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

   if -inf.0 < (*.f64 (*.f64 x y) y) < 3.9999999999999998e-23

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. lift--.f64 N/A

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

       2. flip-- N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-unsound-+.f32 N/A

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

       10. lower-+.f32 N/A

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

       11. metadata-eval N/A

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

       12. sub-flip N/A

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

       13. lift--.f64 N/A

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

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

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

   11. Applied rewrites 57.4%

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

   if 3.9999999999999998e-23 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

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

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      6. lower-unsound-*.f64 67.6%

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      7. lift-pow.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 67.6%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

      12. lower-*.f64 67.6%

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

   6. Applied rewrites 67.6%

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

      1. rem-square-sqrt N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      4. lower-*.f64 67.3%

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      7. associate-*l* N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      11. lower-*.f64 67.3%

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      14. associate-*l* N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)}} \]

      17. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]

      18. lower-*.f64 67.3%

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]

   8. Applied rewrites 67.3%

      \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot y\right) \cdot y\\ t_1 := \left(y \cdot y\right) \cdot x\\ t_2 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -500000000:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\left(y \cdot y\right) \cdot y\right) \cdot y} \cdot x - 1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{t\_1 \cdot t\_1 - 1 \cdot 1}{t\_2 - 1}\\ \mathbf{else}:\\ \;\;\;\;1 + t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* x y) y)) (t_1 (* (* y y) x)) (t_2 (* (* y x) y)))
  (if (<= t_0 -500000000.0)
    (/ -1.0 (- (* (sqrt (* (* (* y y) y) y)) x) 1.0))
    (if (<= t_0 5e+263)
      (/ (- (* t_1 t_1) (* 1.0 1.0)) (- t_2 1.0))
      (+ 1.0 t_2)))))

C

double code(double x, double y) {
	double t_0 = (x * y) * y;
	double t_1 = (y * y) * x;
	double t_2 = (y * x) * y;
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = -1.0 / ((sqrt((((y * y) * y) * y)) * x) - 1.0);
	} else if (t_0 <= 5e+263) {
		tmp = ((t_1 * t_1) - (1.0 * 1.0)) / (t_2 - 1.0);
	} else {
		tmp = 1.0 + t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * y) * y
    t_1 = (y * y) * x
    t_2 = (y * x) * y
    if (t_0 <= (-500000000.0d0)) then
        tmp = (-1.0d0) / ((sqrt((((y * y) * y) * y)) * x) - 1.0d0)
    else if (t_0 <= 5d+263) then
        tmp = ((t_1 * t_1) - (1.0d0 * 1.0d0)) / (t_2 - 1.0d0)
    else
        tmp = 1.0d0 + t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * y) * y;
	double t_1 = (y * y) * x;
	double t_2 = (y * x) * y;
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = -1.0 / ((Math.sqrt((((y * y) * y) * y)) * x) - 1.0);
	} else if (t_0 <= 5e+263) {
		tmp = ((t_1 * t_1) - (1.0 * 1.0)) / (t_2 - 1.0);
	} else {
		tmp = 1.0 + t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) * y
	t_1 = (y * y) * x
	t_2 = (y * x) * y
	tmp = 0
	if t_0 <= -500000000.0:
		tmp = -1.0 / ((math.sqrt((((y * y) * y) * y)) * x) - 1.0)
	elif t_0 <= 5e+263:
		tmp = ((t_1 * t_1) - (1.0 * 1.0)) / (t_2 - 1.0)
	else:
		tmp = 1.0 + t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * y)
	t_1 = Float64(Float64(y * y) * x)
	t_2 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -500000000.0)
		tmp = Float64(-1.0 / Float64(Float64(sqrt(Float64(Float64(Float64(y * y) * y) * y)) * x) - 1.0));
	elseif (t_0 <= 5e+263)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(1.0 * 1.0)) / Float64(t_2 - 1.0));
	else
		tmp = Float64(1.0 + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) * y;
	t_1 = (y * y) * x;
	t_2 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -500000000.0)
		tmp = -1.0 / ((sqrt((((y * y) * y) * y)) * x) - 1.0);
	elseif (t_0 <= 5e+263)
		tmp = ((t_1 * t_1) - (1.0 * 1.0)) / (t_2 - 1.0);
	else
		tmp = 1.0 + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -500000000.0], N[(-1.0 / N[(N[(N[Sqrt[N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+263], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e8

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. rem-square-sqrt N/A

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

       8. sqrt-unprod N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 67.4%

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

       12. lift-sqrt.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. sqrt-unprod N/A

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

       15. rem-square-sqrt 68.0%

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

   11. Applied rewrites 68.0%

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

   if -5e8 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e263

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 5.0000000000000002e263 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

Alternative 3: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sqrt{t\_0} \cdot x - 1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* (* y y) y) y)))
  (if (<= (* (* x y) y) 4e-23)
    (/ -1.0 (- (* (sqrt t_0) x) 1.0))
    (+ 1.0 (* x (sqrt (sqrt (* t_0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = ((y * y) * y) * y;
	double tmp;
	if (((x * y) * y) <= 4e-23) {
		tmp = -1.0 / ((sqrt(t_0) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * sqrt(sqrt((t_0 * 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 = ((y * y) * y) * y
    if (((x * y) * y) <= 4d-23) then
        tmp = (-1.0d0) / ((sqrt(t_0) * x) - 1.0d0)
    else
        tmp = 1.0d0 + (x * sqrt(sqrt((t_0 * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((y * y) * y) * y;
	double tmp;
	if (((x * y) * y) <= 4e-23) {
		tmp = -1.0 / ((Math.sqrt(t_0) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * Math.sqrt(Math.sqrt((t_0 * t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((y * y) * y) * y
	tmp = 0
	if ((x * y) * y) <= 4e-23:
		tmp = -1.0 / ((math.sqrt(t_0) * x) - 1.0)
	else:
		tmp = 1.0 + (x * math.sqrt(math.sqrt((t_0 * t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y * y) * y) * y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 4e-23)
		tmp = Float64(-1.0 / Float64(Float64(sqrt(t_0) * x) - 1.0));
	else
		tmp = Float64(1.0 + Float64(x * sqrt(sqrt(Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((y * y) * y) * y;
	tmp = 0.0;
	if (((x * y) * y) <= 4e-23)
		tmp = -1.0 / ((sqrt(t_0) * x) - 1.0);
	else
		tmp = 1.0 + (x * sqrt(sqrt((t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 4e-23], N[(-1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 3.9999999999999998e-23

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. rem-square-sqrt N/A

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

       8. sqrt-unprod N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 67.4%

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

       12. lift-sqrt.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. sqrt-unprod N/A

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

       15. rem-square-sqrt 68.0%

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

   11. Applied rewrites 68.0%

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

   if 3.9999999999999998e-23 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

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

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      6. lower-unsound-*.f64 67.6%

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      7. lift-pow.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 67.6%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

      12. lower-*.f64 67.6%

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

   6. Applied rewrites 67.6%

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

      1. rem-square-sqrt N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      4. lower-*.f64 67.3%

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      7. associate-*l* N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      11. lower-*.f64 67.3%

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

      14. associate-*l* N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)}} \]

      17. *-commutative N/A

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]

      18. lower-*.f64 67.3%

          \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]

   8. Applied rewrites 67.3%

      \[\leadsto 1 + x \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\left(y \cdot y\right) \cdot y\right) \cdot y} \cdot x - 1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (* (* x y) y) 4e-23)
  (/ -1.0 (- (* (sqrt (* (* (* y y) y) y)) x) 1.0))
  (+ 1.0 (* x (sqrt (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 4e-23) {
		tmp = -1.0 / ((sqrt((((y * y) * y) * y)) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * sqrt(((y * y) * (y * y))));
	}
	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 (((x * y) * y) <= 4d-23) then
        tmp = (-1.0d0) / ((sqrt((((y * y) * y) * y)) * x) - 1.0d0)
    else
        tmp = 1.0d0 + (x * sqrt(((y * y) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 4e-23) {
		tmp = -1.0 / ((Math.sqrt((((y * y) * y) * y)) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * Math.sqrt(((y * y) * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * y) * y) <= 4e-23:
		tmp = -1.0 / ((math.sqrt((((y * y) * y) * y)) * x) - 1.0)
	else:
		tmp = 1.0 + (x * math.sqrt(((y * y) * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 4e-23)
		tmp = Float64(-1.0 / Float64(Float64(sqrt(Float64(Float64(Float64(y * y) * y) * y)) * x) - 1.0));
	else
		tmp = Float64(1.0 + Float64(x * sqrt(Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * y) * y) <= 4e-23)
		tmp = -1.0 / ((sqrt((((y * y) * y) * y)) * x) - 1.0);
	else
		tmp = 1.0 + (x * sqrt(((y * y) * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 4e-23], N[(-1.0 / N[(N[(N[Sqrt[N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 3.9999999999999998e-23

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. rem-square-sqrt N/A

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

       8. sqrt-unprod N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 67.4%

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

       12. lift-sqrt.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. sqrt-unprod N/A

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

       15. rem-square-sqrt 68.0%

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

   11. Applied rewrites 68.0%

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

   if 3.9999999999999998e-23 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

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

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      6. lower-unsound-*.f64 67.6%

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      7. lift-pow.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 67.6%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

      12. lower-*.f64 67.6%

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

   6. Applied rewrites 67.6%

      \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{\left(x \cdot y\right) \cdot y} \leq 2:\\ \;\;\;\;\frac{-1}{\left(y \cdot y\right) \cdot x - 1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (exp (* (* x y) y)) 2.0)
  (/ -1.0 (- (* (* y y) x) 1.0))
  (+ 1.0 (* x (sqrt (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (exp(((x * y) * y)) <= 2.0) {
		tmp = -1.0 / (((y * y) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * sqrt(((y * y) * (y * y))));
	}
	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 (exp(((x * y) * y)) <= 2.0d0) then
        tmp = (-1.0d0) / (((y * y) * x) - 1.0d0)
    else
        tmp = 1.0d0 + (x * sqrt(((y * y) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.exp(((x * y) * y)) <= 2.0) {
		tmp = -1.0 / (((y * y) * x) - 1.0);
	} else {
		tmp = 1.0 + (x * Math.sqrt(((y * y) * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp(((x * y) * y)) <= 2.0:
		tmp = -1.0 / (((y * y) * x) - 1.0)
	else:
		tmp = 1.0 + (x * math.sqrt(((y * y) * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(Float64(x * y) * y)) <= 2.0)
		tmp = Float64(-1.0 / Float64(Float64(Float64(y * y) * x) - 1.0));
	else
		tmp = Float64(1.0 + Float64(x * sqrt(Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (exp(((x * y) * y)) <= 2.0)
		tmp = -1.0 / (((y * y) * x) - 1.0);
	else
		tmp = 1.0 + (x * sqrt(((y * y) * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(-1.0 / N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 2

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

   if 2 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

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

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      6. lower-unsound-*.f64 67.6%

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      7. lift-pow.f64 N/A

         \[\leadsto 1 + x \cdot \sqrt{{y}^{2} \cdot {y}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 67.6%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

      12. lower-*.f64 67.6%

          \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]

   6. Applied rewrites 67.6%

      \[\leadsto 1 + x \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left(y \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-295}:\\ \;\;\;\;\frac{-1}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - -1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* y y) x)))
  (if (<= x -5.6e-295) (/ -1.0 (- t_0 1.0)) (- t_0 -1.0))))

C

double code(double x, double y) {
	double t_0 = (y * y) * x;
	double tmp;
	if (x <= -5.6e-295) {
		tmp = -1.0 / (t_0 - 1.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * x
    if (x <= (-5.6d-295)) then
        tmp = (-1.0d0) / (t_0 - 1.0d0)
    else
        tmp = t_0 - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * x;
	double tmp;
	if (x <= -5.6e-295) {
		tmp = -1.0 / (t_0 - 1.0);
	} else {
		tmp = t_0 - -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * x
	tmp = 0
	if x <= -5.6e-295:
		tmp = -1.0 / (t_0 - 1.0)
	else:
		tmp = t_0 - -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * x)
	tmp = 0.0
	if (x <= -5.6e-295)
		tmp = Float64(-1.0 / Float64(t_0 - 1.0));
	else
		tmp = Float64(t_0 - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * x;
	tmp = 0.0;
	if (x <= -5.6e-295)
		tmp = -1.0 / (t_0 - 1.0);
	else
		tmp = t_0 - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.6e-295], N[(-1.0 / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999998e-295

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

   if -5.5999999999999998e-295 < x

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. remove-double-neg N/A

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

      5. lower--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.1%

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

   6. Applied rewrites 66.1%

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

Alternative 7: 78.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-295}:\\ \;\;\;\;\frac{-1}{\left(y \cdot x\right) \cdot y - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x - -1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= x -5.6e-295)
  (/ -1.0 (- (* (* y x) y) 1.0))
  (- (* (* y y) x) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e-295) {
		tmp = -1.0 / (((y * x) * y) - 1.0);
	} else {
		tmp = ((y * y) * x) - -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 (x <= (-5.6d-295)) then
        tmp = (-1.0d0) / (((y * x) * y) - 1.0d0)
    else
        tmp = ((y * y) * x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e-295) {
		tmp = -1.0 / (((y * x) * y) - 1.0);
	} else {
		tmp = ((y * y) * x) - -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e-295:
		tmp = -1.0 / (((y * x) * y) - 1.0)
	else:
		tmp = ((y * y) * x) - -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e-295)
		tmp = Float64(-1.0 / Float64(Float64(Float64(y * x) * y) - 1.0));
	else
		tmp = Float64(Float64(Float64(y * y) * x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e-295)
		tmp = -1.0 / (((y * x) * y) - 1.0);
	else
		tmp = ((y * y) * x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e-295], N[(-1.0 / N[(N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999998e-295

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 63.3%

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

   11. Applied rewrites 63.3%

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

   if -5.5999999999999998e-295 < x

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 66.1%

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. remove-double-neg N/A

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

      5. lower--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.1%

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

   6. Applied rewrites 66.1%

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

Alternative 8: 66.1% accurate, 7.9× speedup

Math

\[\left(y \cdot y\right) \cdot x - -1 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]

2. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 66.1%

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

4. Applied rewrites 66.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. remove-double-neg N/A

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

   5. lower--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 66.1%

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

6. Applied rewrites 66.1%

   \[\leadsto \left(y \cdot y\right) \cdot x - \color{blue}{-1} \]
7. Add Preprocessing

Alternative 9: 63.4% accurate, 7.9× speedup

Math

\[1 + \left(y \cdot x\right) \cdot y \]

FPCore

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

C

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

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 + ((y * x) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]

2. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 66.1%

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

4. Applied rewrites 66.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 63.4%

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

6. Applied rewrites 63.4%

   \[\leadsto 1 + \left(y \cdot x\right) \cdot \color{blue}{y} \]
7. Add Preprocessing

Alternative 10: 51.2% accurate, 111.0× speedup

Math

\[1 \]

FPCore

(FPCore (x y)
  :precision binary64
  1.0)

C

double code(double x, double y) {
	return 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
end function

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))