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

Percentage Accurate: 66.5% → 96.4%

Time: 3.5s

Alternatives: 10

Speedup: 1.0×

• 48.2% 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: 96.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 81.7

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

4. Applied rewrites 81.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 98.3

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

6. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]
7. Add Preprocessing

Alternative 2: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \left|z\right| \cdot \frac{\left|z\right|}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 4e+174)
   (fma (/ (/ x y) y) x (* (fabs z) (/ (fabs z) (* t t))))
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 4e+174) {
		tmp = fma(((x / y) / y), x, (fabs(z) * (fabs(z) / (t * t))));
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 4e+174)
		tmp = fma(Float64(Float64(x / y) / y), x, Float64(abs(z) * Float64(abs(z) / Float64(t * t))));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 4e+174], N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x + N[(N[Abs[z], $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.00000000000000028e174

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. sqr-abs-rev N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fabs.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 4.00000000000000028e174 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

Alternative 3: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y} + z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 4e+174)
   (+ (/ (* (/ x y) x) y) (* z (/ z (* t t))))
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 4e+174) {
		tmp = (((x / y) * x) / y) + (z * (z / (t * t)));
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 4e+174)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) / y) + Float64(z * Float64(z / Float64(t * t))));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 4e+174], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision] + N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.00000000000000028e174

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 95.2

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

   6. Applied rewrites 95.2%

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

   if 4.00000000000000028e174 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y} + z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 4e+174)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ x (* y y)) x (* (/ z t) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 4e+174) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 4e+174)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+174], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.00000000000000028e174

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

   if 4.00000000000000028e174 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

Alternative 5: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \frac{z \cdot z}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) INFINITY)
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))
   (fma (/ (/ x y) y) x (/ (* z z) (* t t)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 97.5

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

   6. Applied rewrites 97.5%

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

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.3

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

   8. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 96.5

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

   10. Applied rewrites 96.5%

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

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

   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 89.8

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

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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 81.1

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

   6. Applied rewrites 81.1%

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

Alternative 6: 77.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 INFINITY) (fma (/ x (* y y)) x t_1) (/ (* x x) (* y y)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 98.0

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

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   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. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if +inf.0 < (/.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. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 7: 93.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 97.9

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

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
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. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. pow2 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 96.4

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

6. Applied rewrites 96.4%

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

Alternative 8: 89.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 97.9

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

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
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. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. pow2 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 96.4

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

6. Applied rewrites 96.4%

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

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 90.9

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

8. Applied rewrites 90.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 92.3

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

10. Applied rewrites 92.3%

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

Alternative 9: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 97.9

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

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
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. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. pow2 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 96.4

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

6. Applied rewrites 96.4%

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

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 90.9

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

8. Applied rewrites 90.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l* N/A

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

   5. times-frac N/A

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

   6. sqr-abs-rev N/A

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

   7. pow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-fabs.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 81.6

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

10. Applied rewrites 81.6%

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

Alternative 10: 47.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, 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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. times-frac N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 54.0

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

8. Applied rewrites 54.0%

   \[\leadsto \frac{x \cdot x}{\color{blue}{y} \cdot y} \]
9. 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 2025051 
(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))))