Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Percentage Accurate: 90.8% → 97.7%

Time: 5.3s

Alternatives: 11

Speedup: 0.8×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;x\_m \cdot x\_m - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x\_m}, \frac{z}{x\_m}, \frac{-t}{x\_m \cdot x\_m}\right), 1\right) \cdot x\_m\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 8e-6)
   (- (* x_m x_m) (fma (* (* 4.0 y) z) z (* (* 4.0 y) (- t))))
   (*
    (*
     (fma (* -4.0 y) (fma (/ z x_m) (/ z x_m) (/ (- t) (* x_m x_m))) 1.0)
     x_m)
    x_m)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 8e-6) {
		tmp = (x_m * x_m) - fma(((4.0 * y) * z), z, ((4.0 * y) * -t));
	} else {
		tmp = (fma((-4.0 * y), fma((z / x_m), (z / x_m), (-t / (x_m * x_m))), 1.0) * x_m) * x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 8e-6)
		tmp = Float64(Float64(x_m * x_m) - fma(Float64(Float64(4.0 * y) * z), z, Float64(Float64(4.0 * y) * Float64(-t))));
	else
		tmp = Float64(Float64(fma(Float64(-4.0 * y), fma(Float64(z / x_m), Float64(z / x_m), Float64(Float64(-t) / Float64(x_m * x_m))), 1.0) * x_m) * x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 8e-6], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(N[(N[(4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[(4.0 * y), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * y), $MachinePrecision] * N[(N[(z / x$95$m), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision] + N[((-t) / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999964e-6

   1. Initial program 94.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      7. negate-sub N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \color{blue}{\left({z}^{2} + \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \left({z}^{2} + \color{blue}{-1 \cdot t}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto x \cdot x - \color{blue}{\left(\left(4 \cdot y\right) \cdot {z}^{2} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right)} \]

      10. pow2 N/A

          \[\leadsto x \cdot x - \left(\left(4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto x \cdot x - \left(\color{blue}{\left(\left(4 \cdot y\right) \cdot z\right) \cdot z} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot x - \color{blue}{\mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\color{blue}{\left(4 \cdot y\right) \cdot z}, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\color{blue}{\left(4 \cdot y\right)} \cdot z, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \color{blue}{\left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \color{blue}{\left(4 \cdot y\right)} \cdot \left(-1 \cdot t\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

      18. lower-neg.f64 97.3

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \color{blue}{\left(-t\right)}\right) \]

   3. Applied rewrites 97.3%

      \[\leadsto x \cdot x - \color{blue}{\mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-t\right)\right)} \]

   if 7.99999999999999964e-6 < x

   1. Initial program 87.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right) \cdot \color{blue}{{x}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right) \cdot x\right) \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2} - t}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      6. negate-sub N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2} + \left(\mathsf{neg}\left(t\right)\right)}{x \cdot x}, 1\right) \cdot x\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2} + \left(\mathsf{neg}\left(t\right)\right)}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      8. div-add-rev N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2}}{{x}^{2}} + \frac{\mathsf{neg}\left(t\right)}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2}}{{x}^{2}} + \frac{-1 \cdot t}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      10. associate-*r/ N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{{z}^{2}}{{x}^{2}} + -1 \cdot \frac{t}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z}{{x}^{2}} + -1 \cdot \frac{t}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z \cdot z}{x \cdot x} + -1 \cdot \frac{t}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      13. times-frac N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \frac{z}{x} \cdot \frac{z}{x} + -1 \cdot \frac{t}{{x}^{2}}, 1\right) \cdot x\right) \cdot x \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, -1 \cdot \frac{t}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, -1 \cdot \frac{t}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, -1 \cdot \frac{t}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      17. associate-*r/ N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{-1 \cdot t}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{\mathsf{neg}\left(t\right)}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{\mathsf{neg}\left(t\right)}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      20. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{-t}{{x}^{2}}\right), 1\right) \cdot x\right) \cdot x \]

      21. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{-t}{x \cdot x}\right), 1\right) \cdot x\right) \cdot x \]

      22. lift-*.f64 98.1

          \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{-t}{x \cdot x}\right), 1\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 98.1%

      \[\leadsto \left(\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{x}, \frac{z}{x}, \frac{-t}{x \cdot x}\right), 1\right) \cdot x\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;x\_m \cdot x\_m - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= y 4.1e-69)
   (- (* x_m x_m) (fma (* (* 4.0 y) z) z (* (* 4.0 y) (- t))))
   (fma x_m x_m (* (- t (* z z)) (* 4.0 y)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (y <= 4.1e-69) {
		tmp = (x_m * x_m) - fma(((4.0 * y) * z), z, ((4.0 * y) * -t));
	} else {
		tmp = fma(x_m, x_m, ((t - (z * z)) * (4.0 * y)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (y <= 4.1e-69)
		tmp = Float64(Float64(x_m * x_m) - fma(Float64(Float64(4.0 * y) * z), z, Float64(Float64(4.0 * y) * Float64(-t))));
	else
		tmp = fma(x_m, x_m, Float64(Float64(t - Float64(z * z)) * Float64(4.0 * y)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[y, 4.1e-69], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(N[(N[(4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[(4.0 * y), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * x$95$m + N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0999999999999999e-69

   1. Initial program 91.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      7. negate-sub N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \color{blue}{\left({z}^{2} + \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto x \cdot x - \left(4 \cdot y\right) \cdot \left({z}^{2} + \color{blue}{-1 \cdot t}\right) \]

      9. distribute-lft-in N/A

         \[\leadsto x \cdot x - \color{blue}{\left(\left(4 \cdot y\right) \cdot {z}^{2} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right)} \]

      10. pow2 N/A

          \[\leadsto x \cdot x - \left(\left(4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto x \cdot x - \left(\color{blue}{\left(\left(4 \cdot y\right) \cdot z\right) \cdot z} + \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot x - \color{blue}{\mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\color{blue}{\left(4 \cdot y\right) \cdot z}, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\color{blue}{\left(4 \cdot y\right)} \cdot z, z, \left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \color{blue}{\left(4 \cdot y\right) \cdot \left(-1 \cdot t\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \color{blue}{\left(4 \cdot y\right)} \cdot \left(-1 \cdot t\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

      18. lower-neg.f64 96.9

          \[\leadsto x \cdot x - \mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \color{blue}{\left(-t\right)}\right) \]

   3. Applied rewrites 96.9%

      \[\leadsto x \cdot x - \color{blue}{\mathsf{fma}\left(\left(4 \cdot y\right) \cdot z, z, \left(4 \cdot y\right) \cdot \left(-t\right)\right)} \]

   if 4.0999999999999999e-69 < y

   1. Initial program 89.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 95.3%

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

Alternative 3: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 1.2e+175)
   (fma x_m x_m (* (- t (* z z)) (* 4.0 y)))
   (* (* z (* z y)) -4.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+175) {
		tmp = fma(x_m, x_m, ((t - (z * z)) * (4.0 * y)));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.2e+175)
		tmp = fma(x_m, x_m, Float64(Float64(t - Float64(z * z)) * Float64(4.0 * y)));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 1.2e+175], N[(x$95$m * x$95$m + N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e175

   1. Initial program 92.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   if 1.2e175 < z

   1. Initial program 73.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 79.6

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 91.6

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 91.6%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, t \cdot \left(4 \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(z \cdot z\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 1.5e+19)
   (fma x_m x_m (* t (* 4.0 y)))
   (if (<= z 1.2e+175)
     (fma x_m x_m (* (* (* z z) y) -4.0))
     (* (* z (* z y)) -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.5e+19) {
		tmp = fma(x_m, x_m, (t * (4.0 * y)));
	} else if (z <= 1.2e+175) {
		tmp = fma(x_m, x_m, (((z * z) * y) * -4.0));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.5e+19)
		tmp = fma(x_m, x_m, Float64(t * Float64(4.0 * y)));
	elseif (z <= 1.2e+175)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(z * z) * y) * -4.0));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 1.5e+19], N[(x$95$m * x$95$m + N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e+175], N[(x$95$m * x$95$m + N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.5e19

   1. Initial program 93.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{t} \cdot \left(4 \cdot y\right)\right) \]

   if 1.5e19 < z < 1.2e175

   1. Initial program 89.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left({z}^{2} \cdot \color{blue}{y}\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(\left(z \cdot z\right) \cdot y\right) \cdot \color{blue}{-4}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(\left(z \cdot z\right) \cdot y\right) \cdot -4\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \left(\left(z \cdot z\right) \cdot y\right) \cdot -4\right) \]

      6. lift-*.f64 80.6

         \[\leadsto \mathsf{fma}\left(x, x, \left(\left(z \cdot z\right) \cdot y\right) \cdot \color{blue}{-4}\right) \]

   6. Applied rewrites 80.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4}\right) \]

   if 1.2e175 < z

   1. Initial program 73.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 79.6

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 91.6

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 91.6%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, t \cdot \left(4 \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 1.5e+19)
   (fma x_m x_m (* t (* 4.0 y)))
   (if (<= z 1.18e+151)
     (fma (* (* z z) y) -4.0 (* x_m x_m))
     (* (* z (* z y)) -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.5e+19) {
		tmp = fma(x_m, x_m, (t * (4.0 * y)));
	} else if (z <= 1.18e+151) {
		tmp = fma(((z * z) * y), -4.0, (x_m * x_m));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.5e+19)
		tmp = fma(x_m, x_m, Float64(t * Float64(4.0 * y)));
	elseif (z <= 1.18e+151)
		tmp = fma(Float64(Float64(z * z) * y), -4.0, Float64(x_m * x_m));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 1.5e+19], N[(x$95$m * x$95$m + N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e+151], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.5e19

   1. Initial program 93.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{t} \cdot \left(4 \cdot y\right)\right) \]

   if 1.5e19 < z < 1.18000000000000005e151

   1. Initial program 95.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot {z}^{2}\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + -4 \cdot \left(\color{blue}{y} \cdot {z}^{2}\right) \]

      3. +-commutative N/A

         \[\leadsto -4 \cdot \left(y \cdot {z}^{2}\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot -4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot {z}^{2}, \color{blue}{-4}, {x}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({z}^{2} \cdot y, -4, {x}^{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({z}^{2} \cdot y, -4, {x}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, {x}^{2}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, {x}^{2}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right) \]

      11. lift-*.f64 83.0

          \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right) \]

   4. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)} \]

   if 1.18000000000000005e151 < z

   1. Initial program 71.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 76.4

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 89.6

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 89.6%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, t \cdot \left(4 \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 5.5e+14)
   (fma x_m x_m (* t (* 4.0 y)))
   (if (<= z 1.2e+175) (* (- t (* z z)) (* 4.0 y)) (* (* z (* z y)) -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 5.5e+14) {
		tmp = fma(x_m, x_m, (t * (4.0 * y)));
	} else if (z <= 1.2e+175) {
		tmp = (t - (z * z)) * (4.0 * y);
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 5.5e+14)
		tmp = fma(x_m, x_m, Float64(t * Float64(4.0 * y)));
	elseif (z <= 1.2e+175)
		tmp = Float64(Float64(t - Float64(z * z)) * Float64(4.0 * y));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 5.5e+14], N[(x$95$m * x$95$m + N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e+175], N[(N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.5e14

   1. Initial program 93.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{t} \cdot \left(4 \cdot y\right)\right) \]

   if 5.5e14 < z < 1.2e175

   1. Initial program 90.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(z \cdot z - t\right) \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(z \cdot z - t\right) \cdot \left(\mathsf{neg}\left(4 \cdot y\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{neg}\left(\left(z \cdot z - t\right) \cdot \left(4 \cdot y\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(z \cdot z - t\right)\right)\right) \cdot \color{blue}{\left(4 \cdot y\right)} \]

      8. pow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left({z}^{2} - t\right)\right)\right) \cdot \left(4 \cdot y\right) \]

      9. negate-sub2 N/A

         \[\leadsto \left(t - {z}^{2}\right) \cdot \left(\color{blue}{4} \cdot y\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(t - {z}^{2}\right) \cdot \color{blue}{\left(4 \cdot y\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \left(t - {z}^{2}\right) \cdot \left(\color{blue}{4} \cdot y\right) \]

      12. pow2 N/A

          \[\leadsto \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right) \]

      14. lower-*.f64 67.6

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

   4. Applied rewrites 67.6%

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

   if 1.2e175 < z

   1. Initial program 73.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 79.6

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 91.6

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 91.6%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, t \cdot \left(4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 4.2e+46) (fma x_m x_m (* t (* 4.0 y))) (* (* z (* z y)) -4.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4.2e+46) {
		tmp = fma(x_m, x_m, (t * (4.0 * y)));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 4.2e+46)
		tmp = fma(x_m, x_m, Float64(t * Float64(4.0 * y)));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 4.2e+46], N[(x$95$m * x$95$m + N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.2e46

   1. Initial program 93.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \color{blue}{{x}^{2}} - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

      4. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z - t\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} - t\right) \]

      7. lift--.f64 N/A

         \[\leadsto {x}^{2} - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]

      8. *-commutative N/A

         \[\leadsto {x}^{2} - \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z - t\right) \]

      9. pow2 N/A

         \[\leadsto {x}^{2} - \left(4 \cdot y\right) \cdot \left(\color{blue}{{z}^{2}} - t\right) \]

      10. associate-*r* N/A

          \[\leadsto {x}^{2} - \color{blue}{4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      13. pow2 N/A

          \[\leadsto \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(\color{blue}{z \cdot z} - t\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z - t\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot y\right)\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4 \cdot y\right)\right)}\right) \]

   3. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(t - z \cdot z\right) \cdot \left(4 \cdot y\right)\right)} \]

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{t} \cdot \left(4 \cdot y\right)\right) \]

   if 4.2e46 < z

   1. Initial program 80.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 70.8

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 78.9

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 78.9%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 4.2e+46) (fma (* t y) 4.0 (* x_m x_m)) (* (* z (* z y)) -4.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4.2e+46) {
		tmp = fma((t * y), 4.0, (x_m * x_m));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 4.2e+46)
		tmp = fma(Float64(t * y), 4.0, Float64(x_m * x_m));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 4.2e+46], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.2e46

   1. Initial program 93.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + 4 \cdot \left(\color{blue}{t} \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto 4 \cdot \left(t \cdot y\right) + \color{blue}{{x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot 4 + {\color{blue}{x}}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, \color{blue}{4}, {x}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, x \cdot x\right) \]

      8. lift-*.f64 75.8

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, x \cdot x\right) \]

   4. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)} \]

   if 4.2e46 < z

   1. Initial program 80.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 70.8

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 78.9

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 78.9%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{-180}:\\ \;\;\;\;\left(t \cdot y\right) \cdot 4\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= z 7.8e-180)
   (* (* t y) 4.0)
   (if (<= z 5.3e+14) (* x_m x_m) (* (* z (* z y)) -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 7.8e-180) {
		tmp = (t * y) * 4.0;
	} else if (z <= 5.3e+14) {
		tmp = x_m * x_m;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Fortran

x_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_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 7.8d-180) then
        tmp = (t * y) * 4.0d0
    else if (z <= 5.3d+14) then
        tmp = x_m * x_m
    else
        tmp = (z * (z * y)) * (-4.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 7.8e-180) {
		tmp = (t * y) * 4.0;
	} else if (z <= 5.3e+14) {
		tmp = x_m * x_m;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if z <= 7.8e-180:
		tmp = (t * y) * 4.0
	elif z <= 5.3e+14:
		tmp = x_m * x_m
	else:
		tmp = (z * (z * y)) * -4.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (z <= 7.8e-180)
		tmp = Float64(Float64(t * y) * 4.0);
	elseif (z <= 5.3e+14)
		tmp = Float64(x_m * x_m);
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (z <= 7.8e-180)
		tmp = (t * y) * 4.0;
	elseif (z <= 5.3e+14)
		tmp = x_m * x_m;
	else
		tmp = (z * (z * y)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[z, 7.8e-180], N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[z, 5.3e+14], N[(x$95$m * x$95$m), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.8000000000000005e-180

   1. Initial program 92.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{4} \]

      3. lower-*.f64 35.8

         \[\leadsto \left(t \cdot y\right) \cdot 4 \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

   if 7.8000000000000005e-180 < z < 5.3e14

   1. Initial program 98.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

   if 5.3e14 < z

   1. Initial program 82.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{-4} \]

      3. *-commutative N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({z}^{2} \cdot y\right) \cdot -4 \]

      5. pow2 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      6. lift-*.f64 66.4

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

   4. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(z \cdot z\right) \cdot y\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(y \cdot z\right)\right) \cdot -4 \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

      7. lower-*.f64 73.5

         \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]

   6. Applied rewrites 73.5%

      \[\leadsto \left(z \cdot \left(z \cdot y\right)\right) \cdot -4 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(t \cdot y\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 2e-6) (* (* t y) 4.0) (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e-6) {
		tmp = (t * y) * 4.0;
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Fortran

x_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_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2d-6) then
        tmp = (t * y) * 4.0d0
    else
        tmp = x_m * x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e-6) {
		tmp = (t * y) * 4.0;
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if x_m <= 2e-6:
		tmp = (t * y) * 4.0
	else:
		tmp = x_m * x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e-6)
		tmp = Float64(Float64(t * y) * 4.0);
	else
		tmp = Float64(x_m * x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2e-6)
		tmp = (t * y) * 4.0;
	else
		tmp = x_m * x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 2e-6], N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999991e-6

   1. Initial program 94.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{4} \]

      3. lower-*.f64 49.2

         \[\leadsto \left(t \cdot y\right) \cdot 4 \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

   if 1.99999999999999991e-6 < x

   1. Initial program 87.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

Alternative 11: 41.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

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

C

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

Fortran

x_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_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_m * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 90.8%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 41.2

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

4. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64
  (- (* x x) (* (* y 4.0) (- (* z z) t))))