Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Percentage Accurate: 66.5% → 98.2%

Time: 10.0s

Alternatives: 12

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

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, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

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

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

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, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

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

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{x}{y\_m} \cdot x}{y\_m} + \frac{\frac{z}{t} \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y\_m}}{y\_m}, x, {\left(\frac{z}{t}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 5.5e-195)
   (+ (/ (* (/ x y_m) x) y_m) (/ (* (/ z t) z) t))
   (fma (/ (/ x y_m) y_m) x (pow (/ z t) 2.0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 5.5e-195) {
		tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t);
	} else {
		tmp = fma(((x / y_m) / y_m), x, pow((z / t), 2.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 5.5e-195)
		tmp = Float64(Float64(Float64(Float64(x / y_m) * x) / y_m) + Float64(Float64(Float64(z / t) * z) / t));
	else
		tmp = fma(Float64(Float64(x / y_m) / y_m), x, (Float64(z / t) ^ 2.0));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 5.5e-195], N[(N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000003e-195

   1. Initial program 71.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      8. lower-/.f64 80.8

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}} \cdot z}{t} \]

   4. Applied rewrites 80.8%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{\frac{z}{t} \cdot z}{t} \]

      3. pow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot y} + \frac{\frac{z}{t} \cdot z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\color{blue}{y \cdot y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{2}}{y}}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{2}}{y}}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      10. lift-/.f64 96.4

          \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

   6. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

   if 5.5000000000000003e-195 < y

   1. Initial program 68.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 98.7

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y\_m \cdot y\_m}\\ \mathbf{if}\;t\_1 + \frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;t\_1 + \frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{y\_m} \cdot x\right) \cdot t}{t \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y_m y_m))))
   (if (<= (+ t_1 (/ (* z z) (* t t))) INFINITY)
     (+ t_1 (* (/ z (* t t)) z))
     (/ (* (* (/ x y_m) x) t) (* t y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if ((t_1 + ((z * z) / (t * t))) <= ((double) INFINITY)) {
		tmp = t_1 + ((z / (t * t)) * z);
	} else {
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if ((t_1 + ((z * z) / (t * t))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + ((z / (t * t)) * z);
	} else {
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	t_1 = (x * x) / (y_m * y_m)
	tmp = 0
	if (t_1 + ((z * z) / (t * t))) <= math.inf:
		tmp = t_1 + ((z / (t * t)) * z)
	else:
		tmp = (((x / y_m) * x) * t) / (t * y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y_m * y_m))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(z * z) / Float64(t * t))) <= Inf)
		tmp = Float64(t_1 + Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = Float64(Float64(Float64(Float64(x / y_m) * x) * t) / Float64(t * y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	t_1 = (x * x) / (y_m * y_m);
	tmp = 0.0;
	if ((t_1 + ((z * z) / (t * t))) <= Inf)
		tmp = t_1 + ((z / (t * t)) * z);
	else
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 (*.f64 x x) (*.f64 y y)) (/.f64 (*.f64 z z) (*.f64 t t))) < +inf.0

   1. Initial program 89.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{{t}^{2}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      9. associate-/r* N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      11. lower-/.f64 93.0

          \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   4. Applied rewrites 93.0%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

      3. associate-/l/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t \cdot t}} \cdot z \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{\color{blue}{{t}^{2}}} \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{{t}^{2}}} \cdot z \]

      6. pow2 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      7. lift-*.f64 92.0

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

   6. Applied rewrites 92.0%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t \cdot t}} \cdot z \]

   if +inf.0 < (+.f64 (/.f64 (*.f64 x x) (*.f64 y y)) (/.f64 (*.f64 z z) (*.f64 t t)))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 88.2

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 81.5

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

   6. Applied rewrites 81.5%

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

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + \frac{\frac{x}{y}}{y} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} \]

      7. frac-add N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 61.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{t \cdot y} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{t \cdot \color{blue}{\frac{{x}^{2}}{y}}}{t \cdot y} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot x}{y} \cdot t}{t \cdot y} \]

       5. associate-*l/ N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 35.7

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

   11. Applied rewrites 35.7%

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

Alternative 3: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y\_m} \cdot x}{y\_m} + \frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 4e-79)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (+ (/ (* (/ x y_m) x) y_m) (/ (* (/ z t) z) t)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 4e-79) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t);
	}
	return tmp;
}

Fortran

y_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y_m, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t * t)
    if (t_1 <= 4d-79) then
        tmp = ((x / y_m) * (x / y_m)) + t_1
    else
        tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 4e-79) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 4e-79:
		tmp = ((x / y_m) * (x / y_m)) + t_1
	else:
		tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 4e-79)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(x / y_m) * x) / y_m) + Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 4e-79)
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	else
		tmp = (((x / y_m) * x) / y_m) + (((z / t) * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-79], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision] + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4e-79

   1. Initial program 75.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 97.9

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 4e-79 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      8. lower-/.f64 86.2

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}} \cdot z}{t} \]

   4. Applied rewrites 86.2%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{\frac{z}{t} \cdot z}{t} \]

      3. pow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot y} + \frac{\frac{z}{t} \cdot z}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\color{blue}{y \cdot y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{2}}{y}}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{2}}{y}}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      10. lift-/.f64 96.5

          \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

   6. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y\_m}}{y\_m}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+21)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (fma (/ (/ x y_m) y_m) x (/ (* (/ z t) z) t)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e+21) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = fma(((x / y_m) / y_m), x, (((z / t) * z) / t));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e+21)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = fma(Float64(Float64(x / y_m) / y_m), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+21], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e21

   1. Initial program 77.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 98.0

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 2e21 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 63.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 96.9

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{\frac{\frac{z}{t} \cdot z}{t}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m \cdot y\_m}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+167)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (fma (/ x (* y_m y_m)) x (/ (* (/ z t) z) t)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+167) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = fma((x / (y_m * y_m)), x, (((z / t) * z) / t));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+167)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = fma(Float64(x / Float64(y_m * y_m)), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+167], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999997e167

   1. Initial program 78.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 98.1

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 4.9999999999999997e167 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 60.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 96.6

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 90.1

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

   8. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y \cdot y}}, x, \frac{\frac{z}{t} \cdot z}{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y\_m}}{y\_m}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m \cdot y\_m}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+167)
     (fma (/ (/ x y_m) y_m) x t_1)
     (fma (/ x (* y_m y_m)) x (/ (* (/ z t) z) t)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+167) {
		tmp = fma(((x / y_m) / y_m), x, t_1);
	} else {
		tmp = fma((x / (y_m * y_m)), x, (((z / t) * z) / t));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+167)
		tmp = fma(Float64(Float64(x / y_m) / y_m), x, t_1);
	else
		tmp = fma(Float64(x / Float64(y_m * y_m)), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+167], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999997e167

   1. Initial program 78.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 95.6

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

   if 4.9999999999999997e167 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 60.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 96.6

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 90.1

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

   8. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y \cdot y}}, x, \frac{\frac{z}{t} \cdot z}{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y\_m \cdot y\_m} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\left(z \cdot y\_m\right) \cdot z}{t}}{t \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{y\_m} \cdot x\right) \cdot t}{t \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* x x) (* y_m y_m)) 2e-60)
   (/ (/ (* (* z y_m) z) t) (* t y_m))
   (/ (* (* (/ x y_m) x) t) (* t y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 2e-60) {
		tmp = (((z * y_m) * z) / t) / (t * y_m);
	} else {
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	}
	return tmp;
}

Fortran

y_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y_m, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y_m * y_m)) <= 2d-60) then
        tmp = (((z * y_m) * z) / t) / (t * y_m)
    else
        tmp = (((x / y_m) * x) * t) / (t * y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 2e-60) {
		tmp = (((z * y_m) * z) / t) / (t * y_m);
	} else {
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if ((x * x) / (y_m * y_m)) <= 2e-60:
		tmp = (((z * y_m) * z) / t) / (t * y_m)
	else:
		tmp = (((x / y_m) * x) * t) / (t * y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y_m * y_m)) <= 2e-60)
		tmp = Float64(Float64(Float64(Float64(z * y_m) * z) / t) / Float64(t * y_m));
	else
		tmp = Float64(Float64(Float64(Float64(x / y_m) * x) * t) / Float64(t * y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (((x * x) / (y_m * y_m)) <= 2e-60)
		tmp = (((z * y_m) * z) / t) / (t * y_m);
	else
		tmp = (((x / y_m) * x) * t) / (t * y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], 2e-60], N[(N[(N[(N[(z * y$95$m), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] / N[(t * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-60

   1. Initial program 74.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 98.5

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + \frac{\frac{x}{y}}{y} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} \]

      7. frac-add N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 85.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y \cdot {z}^{2}}{t}}}{t \cdot y} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{y \cdot {z}^{2}}{\color{blue}{t}}}{t \cdot y} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{y \cdot \left(z \cdot z\right)}{t}}{t \cdot y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(y \cdot z\right) \cdot z}{t}}{t \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\left(z \cdot y\right) \cdot z}{t}}{t \cdot y} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(z \cdot y\right) \cdot z}{t}}{t \cdot y} \]

       6. lift-*.f64 73.1

          \[\leadsto \frac{\frac{\left(z \cdot y\right) \cdot z}{t}}{t \cdot y} \]

   11. Applied rewrites 73.1%

       \[\leadsto \frac{\color{blue}{\frac{\left(z \cdot y\right) \cdot z}{t}}}{t \cdot y} \]

   if 1.9999999999999999e-60 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 66.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      8. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      11. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      14. pow2 N/A

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

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      22. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      23. lower-/.f64 94.1

          \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 94.1

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

   6. Applied rewrites 94.1%

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

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + \frac{\frac{x}{y}}{y} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} \]

      7. frac-add N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{t \cdot y} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{t \cdot \color{blue}{\frac{{x}^{2}}{y}}}{t \cdot y} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot x}{y} \cdot t}{t \cdot y} \]

       5. associate-*l/ N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 72.5

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

   11. Applied rewrites 72.5%

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

Alternative 8: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \mathsf{fma}\left(\frac{x}{y\_m \cdot y\_m}, x, \frac{\frac{z}{t} \cdot z}{t}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (fma (/ x (* y_m y_m)) x (/ (* (/ z t) z) t)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	return fma((x / (y_m * y_m)), x, (((z / t) * z) / t));
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return fma(Float64(x / Float64(y_m * y_m)), x, Float64(Float64(Float64(z / t) * z) / t))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

   8. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   14. pow2 N/A

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

   15. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   18. pow2 N/A

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

   19. pow2 N/A

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

   20. times-frac N/A

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

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   22. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   23. lower-/.f64 96.0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   2. pow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-*.f64 93.9

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

6. Applied rewrites 93.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 85.6

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

8. Applied rewrites 85.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y \cdot y}}, x, \frac{\frac{z}{t} \cdot z}{t}\right) \]
9. Add Preprocessing

Alternative 9: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \mathsf{fma}\left(\frac{x}{y\_m \cdot y\_m}, x, z \cdot \frac{z}{t \cdot t}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (fma (/ x (* y_m y_m)) x (* z (/ z (* t t)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	return fma((x / (y_m * y_m)), x, (z * (z / (t * t))));
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return fma(Float64(x / Float64(y_m * y_m)), x, Float64(z * Float64(z / Float64(t * t))))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

   8. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   14. pow2 N/A

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

   15. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   18. pow2 N/A

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

   19. pow2 N/A

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

   20. times-frac N/A

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

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   22. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   23. lower-/.f64 96.0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   2. pow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. sqr-neg-rev N/A

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

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \frac{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)}{\color{blue}{{t}^{2}}}\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{\mathsf{neg}\left(z\right)}{{t}^{2}}}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{\mathsf{neg}\left(z\right)}{{t}^{2}}}\right) \]

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 86.5

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

6. Applied rewrites 86.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 79.6

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

8. Applied rewrites 79.6%

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

9. Final simplification 79.6%

   \[\leadsto \mathsf{fma}\left(\frac{x}{y \cdot y}, x, z \cdot \frac{z}{t \cdot t}\right) \]
10. Add Preprocessing

Alternative 10: 52.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{\left(\frac{x}{y\_m} \cdot x\right) \cdot t}{t \cdot y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (/ (* (* (/ x y_m) x) t) (* t y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	return (((x / y_m) * x) * t) / (t * y_m);
}

Fortran

y_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y_m, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x / y_m) * x) * t) / (t * y_m)
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return (((x / y_m) * x) * t) / (t * y_m);
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return (((x / y_m) * x) * t) / (t * y_m)

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return Float64(Float64(Float64(Float64(x / y_m) * x) * t) / Float64(t * y_m))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z, t)
	tmp = (((x / y_m) * x) * t) / (t * y_m);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

   8. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   14. pow2 N/A

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

   15. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   18. pow2 N/A

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

   19. pow2 N/A

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

   20. times-frac N/A

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

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   22. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   23. lower-/.f64 96.0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   2. pow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-*.f64 93.9

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

6. Applied rewrites 93.9%

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

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + \frac{\frac{x}{y}}{y} \cdot x \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} \]

   7. frac-add N/A

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

   8. associate-*l/ N/A

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

   9. pow2 N/A

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

   10. lower-/.f64 N/A

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

8. Applied rewrites 83.3%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{t \cdot y} \]
10. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \frac{t \cdot \color{blue}{\frac{{x}^{2}}{y}}}{t \cdot y} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{{x}^{2}}{y} \cdot \color{blue}{t}}{t \cdot y} \]

    4. pow2 N/A

       \[\leadsto \frac{\frac{x \cdot x}{y} \cdot t}{t \cdot y} \]

    5. associate-*l/ N/A

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

    6. lift-/.f64 N/A

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

    7. lift-*.f64 53.1

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

11. Applied rewrites 53.1%

    \[\leadsto \frac{\color{blue}{\left(\frac{x}{y} \cdot x\right) \cdot t}}{t \cdot y} \]
12. Add Preprocessing

Alternative 11: 46.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{\left(x \cdot x\right) \cdot t}{\left(t \cdot y\_m\right) \cdot y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (/ (* (* x x) t) (* (* t y_m) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	return ((x * x) * t) / ((t * y_m) * y_m);
}

Fortran

y_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y_m, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) * t) / ((t * y_m) * y_m)
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return ((x * x) * t) / ((t * y_m) * y_m);
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return ((x * x) * t) / ((t * y_m) * y_m)

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return Float64(Float64(Float64(x * x) * t) / Float64(Float64(t * y_m) * y_m))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z, t)
	tmp = ((x * x) * t) / ((t * y_m) * y_m);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] * t), $MachinePrecision] / N[(N[(t * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

   8. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   14. pow2 N/A

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

   15. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   18. pow2 N/A

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

   19. pow2 N/A

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

   20. times-frac N/A

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

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   22. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   23. lower-/.f64 96.0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

   8. *-commutative N/A

      \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \color{blue}{x \cdot \frac{\frac{x}{y}}{y}} \]

   9. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   11. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   12. associate-/l/ N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + x \cdot \color{blue}{\frac{x}{y \cdot y}} \]

   13. pow2 N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + x \cdot \frac{x}{\color{blue}{{y}^{2}}} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{x \cdot x}{{y}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \frac{\color{blue}{{x}^{2}}}{{y}^{2}} \]

   16. frac-add N/A

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

   17. lift-/.f64 N/A

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

   18. associate-*l/ N/A

       \[\leadsto \frac{\color{blue}{\frac{z \cdot z}{t}} \cdot {y}^{2} + t \cdot {x}^{2}}{t \cdot {y}^{2}} \]

6. Applied rewrites 51.1%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 44.5

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

9. Applied rewrites 44.5%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 45.8

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

11. Applied rewrites 45.8%

    \[\leadsto \frac{\left(x \cdot x\right) \cdot t}{\color{blue}{\left(t \cdot y\right) \cdot y}} \]
12. Add Preprocessing

Alternative 12: 44.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{\left(t \cdot x\right) \cdot x}{t \cdot \left(y\_m \cdot y\_m\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (/ (* (* t x) x) (* t (* y_m y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	return ((t * x) * x) / (t * (y_m * y_m));
}

Fortran

y_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y_m, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((t * x) * x) / (t * (y_m * y_m))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return ((t * x) * x) / (t * (y_m * y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return ((t * x) * x) / (t * (y_m * y_m))

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return Float64(Float64(Float64(t * x) * x) / Float64(t * Float64(y_m * y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z, t)
	tmp = ((t * x) * x) / (t * (y_m * y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(N[(t * x), $MachinePrecision] * x), $MachinePrecision] / N[(t * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{z \cdot z}}{t \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{\color{blue}{t \cdot t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

   8. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{x}{{y}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{x}{{y}^{2}} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{{y}^{2}} \cdot x + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, x, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   14. pow2 N/A

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

   15. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{y}}{y}}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{y}}}{y}, x, \frac{{z}^{2}}{{t}^{2}}\right) \]

   18. pow2 N/A

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

   19. pow2 N/A

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

   20. times-frac N/A

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

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   22. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \color{blue}{{\left(\frac{z}{t}\right)}^{2}}\right) \]

   23. lower-/.f64 96.0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\color{blue}{\left(\frac{z}{t}\right)}}^{2}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{y} \cdot x + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t} + \frac{\frac{x}{y}}{y} \cdot x} \]

   8. *-commutative N/A

      \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \color{blue}{x \cdot \frac{\frac{x}{y}}{y}} \]

   9. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   11. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} + x \cdot \frac{\frac{x}{y}}{y} \]

   12. associate-/l/ N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + x \cdot \color{blue}{\frac{x}{y \cdot y}} \]

   13. pow2 N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + x \cdot \frac{x}{\color{blue}{{y}^{2}}} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \color{blue}{\frac{x \cdot x}{{y}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} + \frac{\color{blue}{{x}^{2}}}{{y}^{2}} \]

   16. frac-add N/A

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

   17. lift-/.f64 N/A

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

   18. associate-*l/ N/A

       \[\leadsto \frac{\color{blue}{\frac{z \cdot z}{t}} \cdot {y}^{2} + t \cdot {x}^{2}}{t \cdot {y}^{2}} \]

6. Applied rewrites 51.1%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 44.5

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

9. Applied rewrites 44.5%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. pow2 N/A

       \[\leadsto \frac{{x}^{2} \cdot t}{t \cdot \left(y \cdot y\right)} \]

    4. *-commutative N/A

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

    5. pow2 N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 45.6

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

11. Applied rewrites 45.6%

    \[\leadsto \frac{\left(t \cdot x\right) \cdot \color{blue}{x}}{t \cdot \left(y \cdot y\right)} \]
12. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return pow((x / y), 2.0) + pow((z / t), 2.0);
}

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, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) ** 2.0d0) + ((z / t) ** 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((x / y), 2.0) + Math.pow((z / t), 2.0);
}

Python

def code(x, y, z, t):
	return math.pow((x / y), 2.0) + math.pow((z / t), 2.0)

Julia

function code(x, y, z, t)
	return Float64((Float64(x / y) ^ 2.0) + (Float64(z / t) ^ 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) ^ 2.0) + ((z / t) ^ 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025043 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (pow (/ x y) 2) (pow (/ z t) 2)))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))