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

Percentage Accurate: 90.6% → 95.0%

Time: 8.1s

Alternatives: 8

Speedup: 0.8×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    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: 90.6% 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

real(8) function code(x, y, z, t)
    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: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(z \cdot y\right), -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+162)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma (* z (* z y)) -4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+162) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma((z * (z * y)), -4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+162)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(Float64(z * Float64(z * y)), -4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+162], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e162

   1. Initial program 97.6%

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

   if 1.9999999999999999e162 < (*.f64 z z)

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   7. Applied rewrites 96.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(z \cdot y\right), -4, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -4e-119)
     (* y (* 4.0 t))
     (if (<= t_1 1e+190) (* x x) (* (* z (* z y)) -4.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -4e-119) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 1e+190) {
		tmp = x * x;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 <= (-4d-119)) then
        tmp = y * (4.0d0 * t)
    else if (t_1 <= 1d+190) then
        tmp = 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 t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -4e-119) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 1e+190) {
		tmp = x * x;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -4e-119:
		tmp = y * (4.0 * t)
	elif t_1 <= 1e+190:
		tmp = x * x
	else:
		tmp = (z * (z * y)) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -4e-119)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (t_1 <= 1e+190)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -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 <= -4e-119)
		tmp = y * (4.0 * t);
	elseif (t_1 <= 1e+190)
		tmp = x * x;
	else
		tmp = (z * (z * y)) * -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, -4e-119], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+190], N[(x * x), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.00000000000000005e-119

   1. Initial program 95.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -4.00000000000000005e-119 < (-.f64 (*.f64 z z) t) < 1.0000000000000001e190

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 1.0000000000000001e190 < (-.f64 (*.f64 z z) t)

   1. Initial program 85.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 72.6

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

   7. Applied rewrites 72.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -4 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \cdot z - t \leq 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -4e-119)
     (* y (* 4.0 t))
     (if (<= t_1 1e+190) (* x x) (* y (* (* z 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 <= -4e-119) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 1e+190) {
		tmp = x * x;
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 <= (-4d-119)) then
        tmp = y * (4.0d0 * t)
    else if (t_1 <= 1d+190) then
        tmp = x * x
    else
        tmp = y * ((z * 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 <= -4e-119) {
		tmp = y * (4.0 * t);
	} else if (t_1 <= 1e+190) {
		tmp = x * x;
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -4e-119:
		tmp = y * (4.0 * t)
	elif t_1 <= 1e+190:
		tmp = x * x
	else:
		tmp = y * ((z * 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 <= -4e-119)
		tmp = Float64(y * Float64(4.0 * t));
	elseif (t_1 <= 1e+190)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(Float64(z * 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 <= -4e-119)
		tmp = y * (4.0 * t);
	elseif (t_1 <= 1e+190)
		tmp = x * x;
	else
		tmp = y * ((z * 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, -4e-119], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+190], N[(x * x), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.00000000000000005e-119

   1. Initial program 95.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -4.00000000000000005e-119 < (-.f64 (*.f64 z z) t) < 1.0000000000000001e190

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 1.0000000000000001e190 < (-.f64 (*.f64 z z) t)

   1. Initial program 85.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -4 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;z \cdot z - t \leq 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1500000000:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot z\right) \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1500000000.0)
   (fma y (* 4.0 t) (* x x))
   (if (<= z 6.4e+160)
     (fma y (* (* z z) -4.0) (* x x))
     (* (* z (* z y)) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1500000000.0) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else if (z <= 6.4e+160) {
		tmp = fma(y, ((z * z) * -4.0), (x * x));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1500000000.0)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	elseif (z <= 6.4e+160)
		tmp = fma(y, Float64(Float64(z * z) * -4.0), Float64(x * x));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1500000000.0], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.4e+160], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.5e9

   1. Initial program 93.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. cancel-sign-sub-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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 1.5e9 < z < 6.3999999999999995e160

   1. Initial program 96.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. cancel-sign-sub-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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 6.3999999999999995e160 < z

   1. Initial program 82.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1500000000:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot z\right) \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(z \cdot y\right), -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+17)
   (fma y (* 4.0 t) (* x x))
   (fma (* z (* z y)) -4.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+17) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else {
		tmp = fma((z * (z * y)), -4.0, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+17)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	else
		tmp = fma(Float64(z * Float64(z * y)), -4.0, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+17], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e17

   1. Initial program 97.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. cancel-sign-sub-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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if 5e17 < (*.f64 z z)

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   7. Applied rewrites 91.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(z \cdot y\right), -4, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.4e+96) (fma y (* 4.0 t) (* x x)) (* (* z (* z y)) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.4e+96) {
		tmp = fma(y, (4.0 * t), (x * x));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.4e+96)
		tmp = fma(y, Float64(4.0 * t), Float64(x * x));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2.4e+96], N[(y * N[(4.0 * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.39999999999999993e96

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if 2.39999999999999993e96 < z

   1. Initial program 86.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.2e+77) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.2e+77) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 <= 2.2d+77) then
        tmp = y * (4.0d0 * t)
    else
        tmp = 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 <= 2.2e+77) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.2e+77:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.2e+77)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.2e+77)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.2e+77], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e77

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   if 2.2e77 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

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

real(8) function code(x, y, z, t)
    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(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 92.7%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 39.0

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

5. Applied rewrites 39.0%

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

Developer Target 1: 90.6% 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

real(8) function code(x, y, z, t)
    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 2024216 
(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))))