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

Percentage Accurate: 66.2% → 99.7%

Time: 4.4s

Alternatives: 14

Speedup: 0.8×

• 46.9% 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: 66.2% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, z \cdot \frac{-1}{-t}, {\left(\frac{x}{y}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ z t) (* z (/ -1.0 (- t))) (pow (/ x y) 2.0)))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), (z * (-1.0 / -t)), pow((x / y), 2.0));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), Float64(z * Float64(-1.0 / Float64(-t))), (Float64(x / y) ^ 2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z * N[(-1.0 / (-t)), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

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

   9. times-frac N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. times-frac N/A

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

   18. pow2 N/A

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

   19. lower-pow.f64 N/A

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

   20. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{z \cdot \frac{-1}{-t}}, {\left(\frac{x}{y}\right)}^{2}\right) \]
7. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{-z}{t \cdot t}, {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e+104)
   (fma (- z) (/ (- z) (* t t)) (pow (/ x y) 2.0))
   (fma (/ z t) (/ z t) (/ (* (/ x y) x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e+104) {
		tmp = fma(-z, (-z / (t * t)), pow((x / y), 2.0));
	} else {
		tmp = fma((z / t), (z / t), (((x / y) * x) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e+104)
		tmp = fma(Float64(-z), Float64(Float64(-z) / Float64(t * t)), (Float64(x / y) ^ 2.0));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y) * x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e+104], N[((-z) * N[((-z) / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e104

   1. Initial program 73.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. times-frac N/A

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

      23. pow2 N/A

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

      24. lower-pow.f64 N/A

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

      25. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 2e104 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 98.4

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

   6. Applied rewrites 98.4%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ z t) (/ z t) (pow (/ x y) 2.0)))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), (z / t), pow((x / y), 2.0));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), Float64(z / t), (Float64(x / y) ^ 2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

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

   9. times-frac N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. times-frac N/A

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

   18. pow2 N/A

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

   19. lower-pow.f64 N/A

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

   20. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(-x\right) \cdot \frac{-x}{y \cdot y} + \frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 4e-100)
     (/ (* (/ x y) x) y)
     (if (<= t_1 INFINITY)
       (+ (* (- x) (/ (- x) (* y y))) (* (/ z (* t t)) z))
       (fma (/ z t) (/ z t) (/ (* 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 <= 4e-100) {
		tmp = ((x / y) * x) / y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z);
	} else {
		tmp = fma((z / t), (z / t), ((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 <= 4e-100)
		tmp = Float64(Float64(Float64(x / y) * x) / y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(-x) / Float64(y * y))) + Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), 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, 4e-100], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[((-x) * N[((-x) / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.0000000000000001e-100

   1. Initial program 72.2%

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

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 25.0

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

   7. Applied rewrites 25.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 26.3

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

   9. Applied rewrites 26.3%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 85.2

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

   12. Applied rewrites 85.2%

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

   if 4.0000000000000001e-100 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.2%

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

      1. lift-*.f64 N/A

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

      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. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.3

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

   4. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 91.5

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

   8. Applied rewrites 91.5%

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

   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. 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. +-commutative N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 5: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (/ (* (/ x y) x) y)))
   (if (<= t_1 4e-100)
     t_2
     (if (<= t_1 INFINITY) (+ (* (/ x (* y y)) x) t_1) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 4e-100) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x / (y * y)) * x) + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 4e-100) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / (y * y)) * x) + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = ((x / y) * x) / y
	tmp = 0
	if t_1 <= 4e-100:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = ((x / (y * y)) * x) + t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(Float64(x / y) * x) / y)
	tmp = 0.0
	if (t_1 <= 4e-100)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x / Float64(y * y)) * x) + t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = ((x / y) * x) / y;
	tmp = 0.0;
	if (t_1 <= 4e-100)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = ((x / (y * y)) * x) + t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-100], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.0000000000000001e-100 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.5%

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

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.2

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

   7. Applied rewrites 19.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 20.2

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

   9. Applied rewrites 20.2%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if 4.0000000000000001e-100 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.2%

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

      1. lift-*.f64 N/A

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

      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. pow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 89.3

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

   4. Applied rewrites 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 87.3

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

   6. Applied rewrites 87.3%

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

Alternative 6: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (/ (* (/ x y) x) y)))
   (if (<= t_1 1e-81)
     t_2
     (if (<= t_1 INFINITY) (/ (/ (* (* z z) y) t) (* y t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((z * z) * y) / t) / (y * t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((z * z) * y) / t) / (y * t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = ((x / y) * x) / y
	tmp = 0
	if t_1 <= 1e-81:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (((z * z) * y) / t) / (y * t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(Float64(x / y) * x) / y)
	tmp = 0.0
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(z * z) * y) / t) / Float64(y * t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = ((x / y) * x) / y;
	tmp = 0.0;
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (((z * z) * y) / t) / (y * t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-81], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision] / N[(y * t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999996e-82 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.3

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

   7. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 20.4

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 75.7

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

   12. Applied rewrites 75.7%

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

   if 9.9999999999999996e-82 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.2%

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

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.2

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

   9. Applied rewrites 74.2%

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

Alternative 7: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(y \cdot z\right) \cdot z}{\left(y \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (/ (* (/ x y) x) y)))
   (if (<= t_1 1e-81)
     t_2
     (if (<= t_1 INFINITY) (/ (* (* y z) z) (* (* y t) t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((y * z) * z) / ((y * t) * t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((y * z) * z) / ((y * t) * t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = ((x / y) * x) / y
	tmp = 0
	if t_1 <= 1e-81:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = ((y * z) * z) / ((y * t) * t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(Float64(x / y) * x) / y)
	tmp = 0.0
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(y * z) * z) / Float64(Float64(y * t) * t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = ((x / y) * x) / y;
	tmp = 0.0;
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = ((y * z) * z) / ((y * t) * t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-81], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] / N[(N[(y * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999996e-82 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.3

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

   7. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 20.4

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 75.7

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

   12. Applied rewrites 75.7%

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

   if 9.9999999999999996e-82 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.2%

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

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 73.3

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

   11. Applied rewrites 73.3%

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

Alternative 8: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(y \cdot z\right) \cdot z}{\left(t \cdot t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (/ (* (/ x y) x) y)))
   (if (<= t_1 1e-81)
     t_2
     (if (<= t_1 INFINITY) (/ (* (* y z) z) (* (* t t) y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((y * z) * z) / ((t * t) * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = ((x / y) * x) / y;
	double tmp;
	if (t_1 <= 1e-81) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((y * z) * z) / ((t * t) * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = ((x / y) * x) / y
	tmp = 0
	if t_1 <= 1e-81:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = ((y * z) * z) / ((t * t) * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(Float64(x / y) * x) / y)
	tmp = 0.0
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(y * z) * z) / Float64(Float64(t * t) * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = ((x / y) * x) / y;
	tmp = 0.0;
	if (t_1 <= 1e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = ((y * z) * z) / ((t * t) * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-81], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] / N[(N[(t * t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999996e-82 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.3

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

   7. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 20.4

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 75.7

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

   12. Applied rewrites 75.7%

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

   if 9.9999999999999996e-82 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.2%

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

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

Alternative 9: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|x\right| \cdot \frac{\left|x\right|}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{-z}{t \cdot t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e+250)
   (fma (/ z t) (/ z t) (* (fabs x) (/ (fabs x) (* y y))))
   (fma (- z) (/ (- z) (* t t)) (* (/ (/ x y) y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+250) {
		tmp = fma((z / t), (z / t), (fabs(x) * (fabs(x) / (y * y))));
	} else {
		tmp = fma(-z, (-z / (t * t)), (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e+250)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(abs(x) * Float64(abs(x) / Float64(y * y))));
	else
		tmp = fma(Float64(-z), Float64(Float64(-z) / Float64(t * t)), Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e+250], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-z) * N[((-z) / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999999999999992e249

   1. Initial program 73.0%

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

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. sqr-abs-rev N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

   if 9.9999999999999992e249 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. times-frac N/A

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

      23. pow2 N/A

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

      24. lower-pow.f64 N/A

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

      25. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 90.4

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

   6. Applied rewrites 90.4%

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

Alternative 10: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{-z}{t \cdot t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 1e+250)
     (fma (/ z t) (/ z t) t_1)
     (fma (- z) (/ (- z) (* t t)) (* (/ (/ x y) y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e+250) {
		tmp = fma((z / t), (z / t), t_1);
	} else {
		tmp = fma(-z, (-z / (t * t)), (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 1e+250)
		tmp = fma(Float64(z / t), Float64(z / t), t_1);
	else
		tmp = fma(Float64(-z), Float64(Float64(-z) / Float64(t * t)), Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+250], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[((-z) * N[((-z) / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999999999999992e249

   1. Initial program 73.0%

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

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 9.9999999999999992e249 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. times-frac N/A

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

      23. pow2 N/A

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

      24. lower-pow.f64 N/A

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

      25. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 90.4

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

   6. Applied rewrites 90.4%

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

Alternative 11: 89.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] + 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 75.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 92.4

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

   6. Applied rewrites 92.4%

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

   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

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

      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. pow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 70.0

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

   4. Applied rewrites 70.0%

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

Alternative 12: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{-z}{t \cdot t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.6e+225)
   (fma (/ z t) (/ z t) (/ (* (/ x y) x) y))
   (fma (- z) (/ (- z) (* t t)) (* (/ (/ x y) y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.6e+225) {
		tmp = fma((z / t), (z / t), (((x / y) * x) / y));
	} else {
		tmp = fma(-z, (-z / (t * t)), (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.6e+225)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y) * x) / y));
	else
		tmp = fma(Float64(-z), Float64(Float64(-z) / Float64(t * t)), Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.6e+225], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[((-z) * N[((-z) / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.59999999999999995e225

   1. Initial program 66.5%

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

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

      17. times-frac N/A

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

      18. pow2 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 1.59999999999999995e225 < x

   1. Initial program 62.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. times-frac N/A

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

      23. pow2 N/A

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

      24. lower-pow.f64 N/A

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

      25. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 94.8

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

   6. Applied rewrites 94.8%

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

Alternative 13: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-290}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{-x}{y \cdot y} + \frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e-290)
   (/ (* (/ x y) x) y)
   (+ (* (- x) (/ (- x) (* y y))) (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-290) {
		tmp = ((x / y) * x) / y;
	} else {
		tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, 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
    real(8) :: tmp
    if ((z * z) <= 1d-290) then
        tmp = ((x / y) * x) / y
    else
        tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-290) {
		tmp = ((x / y) * x) / y;
	} else {
		tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e-290:
		tmp = ((x / y) * x) / y
	else:
		tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e-290)
		tmp = Float64(Float64(Float64(x / y) * x) / y);
	else
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(-x) / Float64(y * y))) + Float64(Float64(z / Float64(t * t)) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e-290)
		tmp = ((x / y) * x) / y;
	else
		tmp = (-x * (-x / (y * y))) + ((z / (t * t)) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-290], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[((-x) * N[((-x) / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e-290

   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. +-commutative N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 20.1

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

   7. Applied rewrites 20.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 20.3

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

   9. Applied rewrites 20.3%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 78.4

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

   12. Applied rewrites 78.4%

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

   if 1.0000000000000001e-290 < (*.f64 z z)

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      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. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 89.2

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

   6. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 83.8

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

   8. Applied rewrites 83.8%

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

Alternative 14: 57.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. associate-/r* N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 43.6

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

7. Applied rewrites 43.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 43.8

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

9. Applied rewrites 43.8%

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

10. Taylor expanded in x around inf

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

    1. pow2 N/A

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

    2. associate-*l/ N/A

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

    3. lift-/.f64 N/A

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

    4. lift-*.f64 57.1

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

12. Applied rewrites 57.1%

    \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} \]
13. 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 2025089 
(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))))