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

Percentage Accurate: 90.9% → 96.7%

Time: 3.1s

Alternatives: 6

Speedup: 0.9×

• 48.7% 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: 90.9% 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: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.25 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z\_m, z\_m, -t\right), -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2.25e+146)
   (fma (fma z_m z_m (- t)) (* -4.0 y) (* x x))
   (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.25e+146) {
		tmp = fma(fma(z_m, z_m, -t), (-4.0 * y), (x * x));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.25e+146)
		tmp = fma(fma(z_m, z_m, Float64(-t)), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2.25e+146], N[(N[(z$95$m * z$95$m + (-t)), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.25000000000000013e146

   1. Initial program 95.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 96.1%

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

   if 2.25000000000000013e146 < z

   1. Initial program 61.2%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 2: 54.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z\_m \cdot z\_m - t\right) \leq -1 \cdot 10^{-239}:\\ \;\;\;\;\left(t \cdot y\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (- (* x x) (* (* y 4.0) (- (* z_m z_m) t))) -1e-239)
   (* (* t y) 4.0)
   (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) - ((y * 4.0) * ((z_m * z_m) - t))) <= -1e-239) {
		tmp = (t * y) * 4.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) - ((y * 4.0d0) * ((z_m * z_m) - t))) <= (-1d-239)) then
        tmp = (t * y) * 4.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) - ((y * 4.0) * ((z_m * z_m) - t))) <= -1e-239) {
		tmp = (t * y) * 4.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((x * x) - ((y * 4.0) * ((z_m * z_m) - t))) <= -1e-239:
		tmp = (t * y) * 4.0
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z_m * z_m) - t))) <= -1e-239)
		tmp = Float64(Float64(t * y) * 4.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (((x * x) - ((y * 4.0) * ((z_m * z_m) - t))) <= -1e-239)
		tmp = (t * y) * 4.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-239], N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < -1.0000000000000001e-239

   1. Initial program 95.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -1.0000000000000001e-239 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

Alternative 3: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot y, t, x \cdot x\right)\\ \mathbf{elif}\;z\_m \leq 2.25 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m \cdot z\_m\right) \cdot y, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 5.5e-7)
   (fma (* 4.0 y) t (* x x))
   (if (<= z_m 2.25e+146)
     (fma (* (* z_m z_m) y) -4.0 (* x x))
     (* (* (* y z_m) z_m) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5.5e-7) {
		tmp = fma((4.0 * y), t, (x * x));
	} else if (z_m <= 2.25e+146) {
		tmp = fma(((z_m * z_m) * y), -4.0, (x * x));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 5.5e-7)
		tmp = fma(Float64(4.0 * y), t, Float64(x * x));
	elseif (z_m <= 2.25e+146)
		tmp = fma(Float64(Float64(z_m * z_m) * y), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 5.5e-7], N[(N[(4.0 * y), $MachinePrecision] * t + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.25e+146], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.5000000000000003e-7

   1. Initial program 94.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 5.5000000000000003e-7 < z < 2.25000000000000013e146

   1. Initial program 99.9%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 2.25000000000000013e146 < z

   1. Initial program 61.2%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 4: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.55 \cdot 10^{-114}:\\ \;\;\;\;\left(t \cdot y\right) \cdot 4\\ \mathbf{elif}\;z\_m \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2.55e-114)
   (* (* t y) 4.0)
   (if (<= z_m 1.75e+44) (* x x) (* (* (* y z_m) z_m) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.55e-114) {
		tmp = (t * y) * 4.0;
	} else if (z_m <= 1.75e+44) {
		tmp = x * x;
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.55d-114) then
        tmp = (t * y) * 4.0d0
    else if (z_m <= 1.75d+44) then
        tmp = x * x
    else
        tmp = ((y * z_m) * z_m) * (-4.0d0)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.55e-114) {
		tmp = (t * y) * 4.0;
	} else if (z_m <= 1.75e+44) {
		tmp = x * x;
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 2.55e-114:
		tmp = (t * y) * 4.0
	elif z_m <= 1.75e+44:
		tmp = x * x
	else:
		tmp = ((y * z_m) * z_m) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.55e-114)
		tmp = Float64(Float64(t * y) * 4.0);
	elseif (z_m <= 1.75e+44)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.55e-114)
		tmp = (t * y) * 4.0;
	elseif (z_m <= 1.75e+44)
		tmp = x * x;
	else
		tmp = ((y * z_m) * z_m) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2.55e-114], N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[z$95$m, 1.75e+44], N[(x * x), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.55e-114

   1. Initial program 93.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 41.1

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

   5. Applied rewrites 41.1%

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

   if 2.55e-114 < z < 1.75e44

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if 1.75e44 < z

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

Alternative 5: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot y, t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.2e+46) (fma (* 4.0 y) t (* x x)) (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.2e+46) {
		tmp = fma((4.0 * y), t, (x * x));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.2e+46)
		tmp = fma(Float64(4.0 * y), t, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.2e+46], N[(N[(4.0 * y), $MachinePrecision] * t + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.1999999999999998e46

   1. Initial program 94.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if 3.1999999999999998e46 < z

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

Alternative 6: 41.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, 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 x around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 39.9

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

5. Applied rewrites 39.9%

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

Developer Target 1: 91.0% 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 2025084 
(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))))