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

Percentage Accurate: 90.0% → 95.3%

Time: 6.7s

Alternatives: 9

Speedup: 0.7×

• 49.4% 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: 90.0% 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: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(t \cdot y\right) \cdot 4, 1, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{t}, z, -1\right) \cdot t, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.3e-21)
   (fma (* (* -4.0 y) z) z (fma (* (* t y) 4.0) 1.0 (* x x)))
   (fma (* -4.0 y) (* (fma (/ z t) z -1.0) t) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e-21) {
		tmp = fma(((-4.0 * y) * z), z, fma(((t * y) * 4.0), 1.0, (x * x)));
	} else {
		tmp = fma((-4.0 * y), (fma((z / t), z, -1.0) * t), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.3e-21)
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, fma(Float64(Float64(t * y) * 4.0), 1.0, Float64(x * x)));
	else
		tmp = fma(Float64(-4.0 * y), Float64(fma(Float64(z / t), z, -1.0) * t), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.3e-21], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision] * 1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * y), $MachinePrecision] * N[(N[(N[(z / t), $MachinePrecision] * z + -1.0), $MachinePrecision] * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.30000000000000009e-21

   1. Initial program 90.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.8

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 97.7%

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

   if 1.30000000000000009e-21 < t

   1. Initial program 93.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(t \cdot y\right) \cdot 4, 1, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot y, \mathsf{fma}\left(\frac{z}{t}, z, -1\right) \cdot t, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+59)
   (fma (* t 4.0) y (* x x))
   (if (<= (* z z) 2e+296)
     (fma (* (* z z) y) -4.0 (* x x))
     (* (* (* z y) z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+59) {
		tmp = fma((t * 4.0), y, (x * x));
	} else if ((z * z) <= 2e+296) {
		tmp = fma(((z * z) * y), -4.0, (x * x));
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+59)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	elseif (Float64(z * z) <= 2e+296)
		tmp = fma(Float64(Float64(z * z) * y), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+59], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+296], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.9999999999999997e59

   1. Initial program 98.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. sqr-neg-rev N/A

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

      16. distribute-lft-neg-out N/A

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

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

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

      18. unpow2 N/A

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

      19. mul-1-neg N/A

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

      20. fp-cancel-sign-sub N/A

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

      21. *-commutative N/A

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

      22. mul-1-neg N/A

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

      23. distribute-lft-neg-out N/A

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

      24. *-commutative N/A

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 93.4%

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

   if 4.9999999999999997e59 < (*.f64 z z) < 1.99999999999999996e296

   1. Initial program 95.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. sqr-neg-rev N/A

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

      16. distribute-lft-neg-out N/A

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

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

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

      18. unpow2 N/A

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

      19. mul-1-neg N/A

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

      20. fp-cancel-sign-sub N/A

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

      21. *-commutative N/A

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

      22. mul-1-neg N/A

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

   5. Applied rewrites 81.7%

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

   if 1.99999999999999996e296 < (*.f64 z z)

   1. Initial program 73.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 36.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 36.4

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

   6. Applied rewrites 36.4%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.9

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

   9. Applied rewrites 79.9%

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

   11. Applied rewrites 94.0%

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

Alternative 3: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 5e+307) (- (* x x) (* (* y 4.0) t_1)) (* (* (* z y) z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (x * x) - ((y * 4.0) * t_1);
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= 5d+307) then
        tmp = (x * x) - ((y * 4.0d0) * t_1)
    else
        tmp = ((z * y) * z) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (x * x) - ((y * 4.0) * t_1);
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= 5e+307:
		tmp = (x * x) - ((y * 4.0) * t_1)
	else:
		tmp = ((z * y) * z) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= 5e+307)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * t_1));
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= 5e+307)
		tmp = (x * x) - ((y * 4.0) * t_1);
	else
		tmp = ((z * y) * z) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+307], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 z z) t) < 5e307

   1. Initial program 97.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 5e307 < (-.f64 (*.f64 z z) t)

   1. Initial program 72.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 35.9

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.2

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

   9. Applied rewrites 79.2%

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

   11. Applied rewrites 93.8%

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

Alternative 4: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+75}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.4e-131)
   (* (* t 4.0) y)
   (if (<= z 1.4e+75) (* 1.0 (* x x)) (* (* (* z y) z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.4e-131) {
		tmp = (t * 4.0) * y;
	} else if (z <= 1.4e+75) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4.4d-131) then
        tmp = (t * 4.0d0) * y
    else if (z <= 1.4d+75) then
        tmp = 1.0d0 * (x * x)
    else
        tmp = ((z * y) * z) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.4e-131) {
		tmp = (t * 4.0) * y;
	} else if (z <= 1.4e+75) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 4.4e-131:
		tmp = (t * 4.0) * y
	elif z <= 1.4e+75:
		tmp = 1.0 * (x * x)
	else:
		tmp = ((z * y) * z) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.4e-131)
		tmp = Float64(Float64(t * 4.0) * y);
	elseif (z <= 1.4e+75)
		tmp = Float64(1.0 * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.4e-131)
		tmp = (t * 4.0) * y;
	elseif (z <= 1.4e+75)
		tmp = 1.0 * (x * x);
	else
		tmp = ((z * y) * z) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 4.4e-131], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.4e+75], N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.3999999999999999e-131

   1. Initial program 92.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

   7. Applied rewrites 40.6%

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

   if 4.3999999999999999e-131 < z < 1.40000000000000006e75

   1. Initial program 97.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 91.8

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

   9. Applied rewrites 91.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 71.2%

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]

   if 1.40000000000000006e75 < z

   1. Initial program 85.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 84.9

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.5

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

   9. Applied rewrites 66.5%

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

   11. Applied rewrites 73.8%

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

Alternative 5: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+75}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.4e-131)
   (* (* t 4.0) y)
   (if (<= z 1.4e+75) (* 1.0 (* x x)) (* (* (* z z) y) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.4e-131) {
		tmp = (t * 4.0) * y;
	} else if (z <= 1.4e+75) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4.4d-131) then
        tmp = (t * 4.0d0) * y
    else if (z <= 1.4d+75) then
        tmp = 1.0d0 * (x * x)
    else
        tmp = ((z * z) * y) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.4e-131) {
		tmp = (t * 4.0) * y;
	} else if (z <= 1.4e+75) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 4.4e-131:
		tmp = (t * 4.0) * y
	elif z <= 1.4e+75:
		tmp = 1.0 * (x * x)
	else:
		tmp = ((z * z) * y) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.4e-131)
		tmp = Float64(Float64(t * 4.0) * y);
	elseif (z <= 1.4e+75)
		tmp = Float64(1.0 * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * z) * y) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.4e-131)
		tmp = (t * 4.0) * y;
	elseif (z <= 1.4e+75)
		tmp = 1.0 * (x * x);
	else
		tmp = ((z * z) * y) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 4.4e-131], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.4e+75], N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.3999999999999999e-131

   1. Initial program 92.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

   7. Applied rewrites 40.6%

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

   if 4.3999999999999999e-131 < z < 1.40000000000000006e75

   1. Initial program 97.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 91.8

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

   9. Applied rewrites 91.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 71.2%

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]

   if 1.40000000000000006e75 < z

   1. Initial program 85.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

Alternative 6: 76.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.8e+137) (fma (* t 4.0) y (* x x)) (* (* (* z y) z) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.8e+137) {
		tmp = fma((t * 4.0), y, (x * x));
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.8e+137)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 6.8e+137], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.79999999999999973e137

   1. Initial program 93.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. sqr-neg-rev N/A

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

      16. distribute-lft-neg-out N/A

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

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

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

      18. unpow2 N/A

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

      19. mul-1-neg N/A

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

      20. fp-cancel-sign-sub N/A

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

      21. *-commutative N/A

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

      22. mul-1-neg N/A

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

      23. distribute-lft-neg-out N/A

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

      24. *-commutative N/A

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 75.3%

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

   if 6.79999999999999973e137 < z

   1. Initial program 77.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 77.0

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

   6. Applied rewrites 77.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 80.1

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

   9. Applied rewrites 80.1%

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

   11. Applied rewrites 91.3%

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

Alternative 7: 45.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.5e-48) (* (* t 4.0) y) (* 1.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.5e-48) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = 1.0 * (x * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 7.5d-48) then
        tmp = (t * 4.0d0) * y
    else
        tmp = 1.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.5e-48) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = 1.0 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 7.5e-48:
		tmp = (t * 4.0) * y
	else:
		tmp = 1.0 * (x * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.5e-48)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(1.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 7.5e-48)
		tmp = (t * 4.0) * y;
	else
		tmp = 1.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.5e-48], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.50000000000000042e-48

   1. Initial program 91.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   7. Applied rewrites 38.9%

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

   if 7.50000000000000042e-48 < x

   1. Initial program 92.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. pow1/2 N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

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

      28. pow-pow N/A

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

   4. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 47.7

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

   6. Applied rewrites 47.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 94.2

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

   9. Applied rewrites 94.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 60.7%

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 31.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 34.6

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

5. Applied rewrites 34.6%

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

7. Applied rewrites 34.6%

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

Alternative 9: 5.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-rgt-identity N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. associate-*l* N/A

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

   10. unpow2 N/A

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

   11. *-rgt-identity N/A

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

   12. mul-1-neg N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. distribute-lft-neg-out N/A

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

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

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

   18. unpow2 N/A

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

   19. mul-1-neg N/A

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

   20. fp-cancel-sign-sub N/A

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

   21. *-commutative N/A

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

   22. mul-1-neg N/A

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

   23. distribute-lft-neg-out N/A

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

   24. *-commutative N/A

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 16.1%

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

10. Taylor expanded in x around -inf

    \[\leadsto {x}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
11. Step-by-step derivation

12. Applied rewrites 4.8%

    \[\leadsto \left(-x\right) \cdot \color{blue}{x} \]
13. Add Preprocessing

Developer Target 1: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((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) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024354 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))