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

Percentage Accurate: 66.3% → 98.1%

Time: 4.1s

Alternatives: 14

Speedup: 0.7×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{x}{y} \cdot \frac{-1}{y}\right) + \frac{z}{t\_m} \cdot \frac{z}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t\_m}}{t\_m}, z, {\left(\frac{x}{y}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 5e-239)
   (+ (* (- x) (* (/ x y) (/ -1.0 y))) (* (/ z t_m) (/ z t_m)))
   (fma (/ (/ z t_m) t_m) z (pow (/ x y) 2.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 5e-239) {
		tmp = (-x * ((x / y) * (-1.0 / y))) + ((z / t_m) * (z / t_m));
	} else {
		tmp = fma(((z / t_m) / t_m), z, pow((x / y), 2.0));
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 5e-239)
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(x / y) * Float64(-1.0 / y))) + Float64(Float64(z / t_m) * Float64(z / t_m)));
	else
		tmp = fma(Float64(Float64(z / t_m) / t_m), z, (Float64(x / y) ^ 2.0));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 5e-239], N[(N[((-x) * N[(N[(x / y), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t$95$m), $MachinePrecision] * N[(z / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * z + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5e-239

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 71.1

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

   3. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{-x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]
   4. 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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 5e-239 < t

   1. Initial program 66.9%

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

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 98.0

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

   3. Applied rewrites 98.0%

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

Alternative 2: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 + z \cdot \frac{z}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (/ (* (/ z t_m) z) t_m)))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 INFINITY) (+ t_1 (* z (/ z (* t_m t_m)))) t_2))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 + (z * (z / (t_m * t_m)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + (z * (z / (t_m * t_m)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * x) / (y * y)
	t_2 = ((z / t_m) * z) / t_m
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_1 + (z * (z / (t_m * t_m)))
	else:
		tmp = t_2
	return tmp

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(t$95$1 + N[(z * N[(z / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 0.0 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 52.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 23.4

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

   6. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

       4. lift-*.f64 78.3

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

   11. Applied rewrites 78.3%

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

   if 0.0 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 90.1

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

   3. Applied rewrites 90.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 3: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \mathbf{if}\;t\_1 \leq 10^{+264}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t\_m}{t\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (/ (* (/ z t_m) z) t_m)))
   (if (<= t_1 1e+264)
     t_2
     (if (<= t_1 INFINITY) (/ (* (* (/ x y) x) t_m) (* t_m y)) t_2))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 1e+264) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 1e+264) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * x) / (y * y)
	t_2 = ((z / t_m) * z) / t_m
	tmp = 0
	if t_1 <= 1e+264:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (((x / y) * x) * t_m) / (t_m * y)
	else:
		tmp = t_2
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(Float64(z / t_m) * z) / t_m)
	tmp = 0.0
	if (t_1 <= 1e+264)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t_m) / Float64(t_m * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * x) / (y * y);
	t_2 = ((z / t_m) * z) / t_m;
	tmp = 0.0;
	if (t_1 <= 1e+264)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+264], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$m * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.00000000000000004e264 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 60.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 27.2

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

   6. Applied rewrites 27.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

       4. lift-*.f64 70.3

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

   11. Applied rewrites 70.3%

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

   if 1.00000000000000004e264 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 90.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 85.0

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

   6. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 87.8

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

   8. Applied rewrites 87.8%

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

Alternative 4: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \mathbf{if}\;t\_1 \leq 10^{+264}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot \frac{x \cdot t\_m}{y}}{t\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (/ (* (/ z t_m) z) t_m)))
   (if (<= t_1 1e+264)
     t_2
     (if (<= t_1 INFINITY) (/ (* x (/ (* x t_m) y)) (* t_m y)) t_2))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 1e+264) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * ((x * t_m) / y)) / (t_m * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((z / t_m) * z) / t_m;
	double tmp;
	if (t_1 <= 1e+264) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * ((x * t_m) / y)) / (t_m * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * x) / (y * y)
	t_2 = ((z / t_m) * z) / t_m
	tmp = 0
	if t_1 <= 1e+264:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x * ((x * t_m) / y)) / (t_m * y)
	else:
		tmp = t_2
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(Float64(z / t_m) * z) / t_m)
	tmp = 0.0
	if (t_1 <= 1e+264)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * Float64(Float64(x * t_m) / y)) / Float64(t_m * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * x) / (y * y);
	t_2 = ((z / t_m) * z) / t_m;
	tmp = 0.0;
	if (t_1 <= 1e+264)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x * ((x * t_m) / y)) / (t_m * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+264], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(N[(x * t$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.00000000000000004e264 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 60.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 27.2

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

   6. Applied rewrites 27.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

       4. lift-*.f64 70.3

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

   11. Applied rewrites 70.3%

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

   if 1.00000000000000004e264 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 90.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 85.0

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

   6. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 85.0

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

   8. Applied rewrites 85.0%

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

Alternative 5: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t\_m \cdot t\_m}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{-x}{y \cdot y} + \frac{z}{t\_m} \cdot \frac{z}{t\_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t_m t_m))))
   (if (<= t_1 5e+165)
     (+ (* (/ x y) (/ x y)) t_1)
     (+ (* (- x) (/ (- x) (* y y))) (* (/ z t_m) (/ z t_m))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (z * z) / (t_m * t_m);
	double tmp;
	if (t_1 <= 5e+165) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = (-x * (-x / (y * y))) + ((z / t_m) * (z / t_m));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t_m * t_m)
    if (t_1 <= 5d+165) then
        tmp = ((x / y) * (x / y)) + t_1
    else
        tmp = (-x * (-x / (y * y))) + ((z / t_m) * (z / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (z * z) / (t_m * t_m);
	double tmp;
	if (t_1 <= 5e+165) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = (-x * (-x / (y * y))) + ((z / t_m) * (z / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (z * z) / (t_m * t_m)
	tmp = 0
	if t_1 <= 5e+165:
		tmp = ((x / y) * (x / y)) + t_1
	else:
		tmp = (-x * (-x / (y * y))) + ((z / t_m) * (z / t_m))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(z * z) / Float64(t_m * t_m))
	tmp = 0.0
	if (t_1 <= 5e+165)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(-x) / Float64(y * y))) + Float64(Float64(z / t_m) * Float64(z / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (z * z) / (t_m * t_m);
	tmp = 0.0;
	if (t_1 <= 5e+165)
		tmp = ((x / y) * (x / y)) + t_1;
	else
		tmp = (-x * (-x / (y * y))) + ((z / t_m) * (z / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+165], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[((-x) * N[((-x) / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t$95$m), $MachinePrecision] * N[(z / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999997e165

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 4.9999999999999997e165 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 67.1

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

   3. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{-x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]
   4. 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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

Alternative 6: 91.0% accurate, 0.6× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= (/ (* z z) (* t_m t_m)) INFINITY)
   (fma (- z) (/ (- z) (* t_m t_m)) (/ (* (/ x y) x) y))
   (+ (* (/ x (* y y)) x) (/ (* (/ z t_m) z) t_m))))

C

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

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[((-z) * N[((-z) / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 96.1

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

   3. Applied rewrites 96.1%

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

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

      5. associate-*r/ N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 93.1

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

   5. Applied rewrites 93.1%

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

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

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 0.0

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

Alternative 7: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t\_m \cdot t\_m}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t_m t_m))))
   (if (<= t_1 5e+165)
     (+ (* (/ x y) (/ x y)) t_1)
     (+ (* (/ x (* y y)) x) (/ (* (/ z t_m) z) t_m)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (z * z) / (t_m * t_m);
	double tmp;
	if (t_1 <= 5e+165) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t_m * t_m)
    if (t_1 <= 5d+165) then
        tmp = ((x / y) * (x / y)) + t_1
    else
        tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (z * z) / (t_m * t_m);
	double tmp;
	if (t_1 <= 5e+165) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (z * z) / (t_m * t_m)
	tmp = 0
	if t_1 <= 5e+165:
		tmp = ((x / y) * (x / y)) + t_1
	else:
		tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(z * z) / Float64(t_m * t_m))
	tmp = 0.0
	if (t_1 <= 5e+165)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = Float64(Float64(Float64(x / Float64(y * y)) * x) + Float64(Float64(Float64(z / t_m) * z) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (z * z) / (t_m * t_m);
	tmp = 0.0;
	if (t_1 <= 5e+165)
		tmp = ((x / y) * (x / y)) + t_1;
	else
		tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+165], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999997e165

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 4.9999999999999997e165 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 67.5

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

   3. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 8: 88.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 96.9

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

   3. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 80.9

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

   3. Applied rewrites 80.9%

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

Alternative 9: 85.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 96.9

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

   3. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

   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. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 17.7

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

   6. Applied rewrites 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 19.2%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

       4. lift-*.f64 50.2

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

   11. Applied rewrites 50.2%

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

Alternative 10: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t\_m}{t\_m \cdot y}\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + \frac{z \cdot z}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= (* y y) 5e-297)
   (/ (* (* (/ x y) x) t_m) (* t_m y))
   (if (<= (* y y) 2e+232)
     (+ (* (/ x (* y y)) x) (/ (* z z) (* t_m t_m)))
     (/ (* (/ z t_m) z) t_m))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if ((y * y) <= 5e-297) {
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	} else if ((y * y) <= 2e+232) {
		tmp = ((x / (y * y)) * x) + ((z * z) / (t_m * t_m));
	} else {
		tmp = ((z / t_m) * z) / t_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((y * y) <= 5d-297) then
        tmp = (((x / y) * x) * t_m) / (t_m * y)
    else if ((y * y) <= 2d+232) then
        tmp = ((x / (y * y)) * x) + ((z * z) / (t_m * t_m))
    else
        tmp = ((z / t_m) * z) / t_m
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if ((y * y) <= 5e-297) {
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	} else if ((y * y) <= 2e+232) {
		tmp = ((x / (y * y)) * x) + ((z * z) / (t_m * t_m));
	} else {
		tmp = ((z / t_m) * z) / t_m;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if (y * y) <= 5e-297:
		tmp = (((x / y) * x) * t_m) / (t_m * y)
	elif (y * y) <= 2e+232:
		tmp = ((x / (y * y)) * x) + ((z * z) / (t_m * t_m))
	else:
		tmp = ((z / t_m) * z) / t_m
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (Float64(y * y) <= 5e-297)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t_m) / Float64(t_m * y));
	elseif (Float64(y * y) <= 2e+232)
		tmp = Float64(Float64(Float64(x / Float64(y * y)) * x) + Float64(Float64(z * z) / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(z / t_m) * z) / t_m);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if ((y * y) <= 5e-297)
		tmp = (((x / y) * x) * t_m) / (t_m * y);
	elseif ((y * y) <= 2e+232)
		tmp = ((x / (y * y)) * x) + ((z * z) / (t_m * t_m));
	else
		tmp = ((z / t_m) * z) / t_m;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-297], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e+232], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 5e-297

   1. Initial program 61.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 91.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 68.1

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

   6. Applied rewrites 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

   if 5e-297 < (*.f64 y y) < 2.00000000000000011e232

   1. Initial program 76.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 80.8

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

   3. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} \cdot x} + \frac{z \cdot z}{t \cdot t} \]
   4. 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. unpow2 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. unpow2 N/A

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

      7. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 2.00000000000000011e232 < (*.f64 y y)

   1. Initial program 56.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 67.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 28.5

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

   6. Applied rewrites 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

       4. lift-*.f64 76.8

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

   11. Applied rewrites 76.8%

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

Alternative 11: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(-x\right) \cdot \left(\frac{x}{y} \cdot \frac{-1}{y}\right) + \frac{z}{t\_m} \cdot \frac{z}{t\_m} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (+ (* (- x) (* (/ x y) (/ -1.0 y))) (* (/ z t_m) (/ z t_m))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	return (-x * ((x / y) * (-1.0 / y))) + ((z / t_m) * (z / t_m));
}

Fortran

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

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

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

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	return (-x * ((x / y) * (-1.0 / y))) + ((z / t_m) * (z / t_m));
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	return (-x * ((x / y) * (-1.0 / y))) + ((z / t_m) * (z / t_m))

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	return Float64(Float64(Float64(-x) * Float64(Float64(x / y) * Float64(-1.0 / y))) + Float64(Float64(z / t_m) * Float64(z / t_m)))
end

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[((-x) * N[(N[(x / y), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t$95$m), $MachinePrecision] * N[(z / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. 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 73.5

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

3. Applied rewrites 73.5%

   \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{-x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]
4. 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

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 89.9

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

5. Applied rewrites 89.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lower-/.f64 96.8

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

7. Applied rewrites 96.8%

   \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{-1}{y}\right)} + \frac{z}{t} \cdot \frac{z}{t} \]
8. Add Preprocessing

Alternative 12: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x + \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t\_m}}{t\_m}, z, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 4e-227)
   (+ (* (/ (/ x y) y) x) (/ (* (/ z t_m) z) t_m))
   (fma (/ (/ z t_m) t_m) z (/ (* (/ x y) x) y))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 4e-227) {
		tmp = (((x / y) / y) * x) + (((z / t_m) * z) / t_m);
	} else {
		tmp = fma(((z / t_m) / t_m), z, (((x / y) * x) / y));
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 4e-227)
		tmp = Float64(Float64(Float64(Float64(x / y) / y) * x) + Float64(Float64(Float64(z / t_m) * z) / t_m));
	else
		tmp = fma(Float64(Float64(z / t_m) / t_m), z, Float64(Float64(Float64(x / y) * x) / y));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 4e-227], N[(N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * z + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999978e-227

   1. Initial program 61.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 70.8

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 3.99999999999999978e-227 < t

   1. Initial program 67.0%

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

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 98.2

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

   3. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 13: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-225}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + \frac{\frac{z}{t\_m} \cdot z}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t\_m}}{t\_m}, z, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 3.8e-225)
   (+ (* (/ x (* y y)) x) (/ (* (/ z t_m) z) t_m))
   (fma (/ (/ z t_m) t_m) z (/ (* (/ x y) x) y))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 3.8e-225) {
		tmp = ((x / (y * y)) * x) + (((z / t_m) * z) / t_m);
	} else {
		tmp = fma(((z / t_m) / t_m), z, (((x / y) * x) / y));
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 3.8e-225)
		tmp = Float64(Float64(Float64(x / Float64(y * y)) * x) + Float64(Float64(Float64(z / t_m) * z) / t_m));
	else
		tmp = fma(Float64(Float64(z / t_m) / t_m), z, Float64(Float64(Float64(x / y) * x) / y));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 3.8e-225], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * z + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.8000000000000003e-225

   1. Initial program 61.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. 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 71.0

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

   3. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 3.8000000000000003e-225 < t

   1. Initial program 67.0%

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

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 98.2

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

   3. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 14: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \frac{\frac{z}{t\_m} \cdot z}{t\_m} \end{array} \]

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(z / t$95$m), $MachinePrecision] * z), $MachinePrecision] / t$95$m), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. 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. associate-/r* N/A

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

   11. pow2 N/A

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

   12. associate-/r* N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

3. Applied rewrites 81.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 47.8

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

6. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 47.3%

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

9. Taylor expanded in x around 0

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

    1. pow2 N/A

       \[\leadsto \frac{\frac{z \cdot z}{t}}{t} \]

    2. associate-*l/ N/A

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

    3. lift-/.f64 N/A

       \[\leadsto \frac{\frac{z}{t} \cdot z}{t} \]

    4. lift-*.f64 56.8

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

11. Applied rewrites 56.8%

    \[\leadsto \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]
12. Add Preprocessing

Developer Target 1: 99.6% 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 2025093 
(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))))