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

Percentage Accurate: 90.8% → 96.4%

Time: 7.7s

Alternatives: 9

Speedup: 0.9×

• 49.2% 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: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z\_m \cdot z\_m - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z\_m \cdot y, z\_m, t \cdot y\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.7e+162)
   (fma x x (* (* (- (* z_m z_m) t) y) -4.0))
   (fma x x (* (fma (* z_m y) z_m (* t y)) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.7e+162) {
		tmp = fma(x, x, ((((z_m * z_m) - t) * y) * -4.0));
	} else {
		tmp = fma(x, x, (fma((z_m * y), z_m, (t * y)) * -4.0));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.7e+162)
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z_m * z_m) - t) * y) * -4.0));
	else
		tmp = fma(x, x, Float64(fma(Float64(z_m * y), z_m, Float64(t * y)) * -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, 1.7e+162], N[(x * x + N[(N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(z$95$m * y), $MachinePrecision] * z$95$m + N[(t * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.70000000000000001e162

   1. Initial program 88.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 93.1

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

   4. Applied rewrites 93.1%

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

   if 1.70000000000000001e162 < z

   1. Initial program 66.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.8

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

   8. Applied rewrites 89.8%

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

Alternative 2: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := z\_m \cdot z\_m - t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+197}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\_m\right) \cdot y\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* z_m z_m) t)))
   (if (<= t_1 -4e-32)
     (* (* 4.0 y) t)
     (if (<= t_1 2e+197) (* x x) (* (* (* -4.0 z_m) y) z_m)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -4e-32) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+197) {
		tmp = x * x;
	} else {
		tmp = ((-4.0 * z_m) * y) * z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) - t
    if (t_1 <= (-4d-32)) then
        tmp = (4.0d0 * y) * t
    else if (t_1 <= 2d+197) then
        tmp = x * x
    else
        tmp = (((-4.0d0) * z_m) * y) * z_m
    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 t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -4e-32) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+197) {
		tmp = x * x;
	} else {
		tmp = ((-4.0 * z_m) * y) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) - t
	tmp = 0
	if t_1 <= -4e-32:
		tmp = (4.0 * y) * t
	elif t_1 <= 2e+197:
		tmp = x * x
	else:
		tmp = ((-4.0 * z_m) * y) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) - t)
	tmp = 0.0
	if (t_1 <= -4e-32)
		tmp = Float64(Float64(4.0 * y) * t);
	elseif (t_1 <= 2e+197)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(-4.0 * z_m) * y) * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) - t;
	tmp = 0.0;
	if (t_1 <= -4e-32)
		tmp = (4.0 * y) * t;
	elseif (t_1 <= 2e+197)
		tmp = x * x;
	else
		tmp = ((-4.0 * z_m) * y) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-32], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+197], N[(x * x), $MachinePrecision], N[(N[(N[(-4.0 * z$95$m), $MachinePrecision] * y), $MachinePrecision] * z$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.00000000000000022e-32

   1. Initial program 89.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -4.00000000000000022e-32 < (-.f64 (*.f64 z z) t) < 1.9999999999999999e197

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.9999999999999999e197 < (-.f64 (*.f64 z z) t)

   1. Initial program 69.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 76.5%

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

Alternative 3: 60.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := z\_m \cdot z\_m - t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+197}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m \cdot z\_m\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (- (* z_m z_m) t)))
   (if (<= t_1 -4e-32)
     (* (* 4.0 y) t)
     (if (<= t_1 2e+197) (* x x) (* (* (* z_m z_m) y) -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -4e-32) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+197) {
		tmp = x * x;
	} else {
		tmp = ((z_m * z_m) * y) * -4.0;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * z_m) - t
    if (t_1 <= (-4d-32)) then
        tmp = (4.0d0 * y) * t
    else if (t_1 <= 2d+197) then
        tmp = x * x
    else
        tmp = ((z_m * z_m) * y) * (-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 t_1 = (z_m * z_m) - t;
	double tmp;
	if (t_1 <= -4e-32) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+197) {
		tmp = x * x;
	} else {
		tmp = ((z_m * z_m) * y) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) - t
	tmp = 0
	if t_1 <= -4e-32:
		tmp = (4.0 * y) * t
	elif t_1 <= 2e+197:
		tmp = x * x
	else:
		tmp = ((z_m * z_m) * y) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) - t)
	tmp = 0.0
	if (t_1 <= -4e-32)
		tmp = Float64(Float64(4.0 * y) * t);
	elseif (t_1 <= 2e+197)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(z_m * z_m) * y) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) - t;
	tmp = 0.0;
	if (t_1 <= -4e-32)
		tmp = (4.0 * y) * t;
	elseif (t_1 <= 2e+197)
		tmp = x * x;
	else
		tmp = ((z_m * z_m) * y) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-32], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+197], N[(x * x), $MachinePrecision], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -4.00000000000000022e-32

   1. Initial program 89.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -4.00000000000000022e-32 < (-.f64 (*.f64 z z) t) < 1.9999999999999999e197

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.9999999999999999e197 < (-.f64 (*.f64 z z) t)

   1. Initial program 69.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \cdot z - t \leq 2 \cdot 10^{+197}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.15 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z\_m \cdot z\_m - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\_m\right) \cdot y\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2.15e+150)
   (fma x x (* (* (- (* z_m z_m) t) y) -4.0))
   (* (* (* -4.0 z_m) y) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.15e+150) {
		tmp = fma(x, x, ((((z_m * z_m) - t) * y) * -4.0));
	} else {
		tmp = ((-4.0 * z_m) * y) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.15e+150)
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z_m * z_m) - t) * y) * -4.0));
	else
		tmp = Float64(Float64(Float64(-4.0 * z_m) * y) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2.15e+150], N[(x * x + N[(N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z$95$m), $MachinePrecision] * y), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.14999999999999999e150

   1. Initial program 87.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 93.0

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

   4. Applied rewrites 93.0%

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

   if 2.14999999999999999e150 < z

   1. Initial program 69.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 90.9%

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

Alternative 5: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \mathsf{fma}\left(x, x, -4 \cdot \mathsf{fma}\left(z\_m \cdot y, z\_m, \left(-t\right) \cdot y\right)\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (fma x x (* -4.0 (fma (* z_m y) z_m (* (- t) y)))))

C

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

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	return fma(x, x, Float64(-4.0 * fma(Float64(z_m * y), z_m, Float64(Float64(-t) * y))))
end

Wolfram

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

Derivation

1. Initial program 86.3%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
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. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 91.8

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

4. Applied rewrites 91.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 96.4

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

6. Applied rewrites 96.4%

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

7. Final simplification 96.4%

   \[\leadsto \mathsf{fma}\left(x, x, -4 \cdot \mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right)\right) \]
8. Add Preprocessing

Alternative 6: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(t \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\_m\right) \cdot y\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e+141)
   (fma x x (* 4.0 (* t y)))
   (* (* (* -4.0 z_m) y) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e+141) {
		tmp = fma(x, x, (4.0 * (t * y)));
	} else {
		tmp = ((-4.0 * z_m) * y) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+141)
		tmp = fma(x, x, Float64(4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(Float64(-4.0 * z_m) * y) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+141], N[(x * x + N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z$95$m), $MachinePrecision] * y), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000002e141

   1. Initial program 95.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

   if 1.00000000000000002e141 < (*.f64 z z)

   1. Initial program 71.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 85.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(t \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\_m\right) \cdot y\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e+141)
   (fma (* t y) 4.0 (* x x))
   (* (* (* -4.0 z_m) y) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e+141) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = ((-4.0 * z_m) * y) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+141)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z_m) * y) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+141], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z$95$m), $MachinePrecision] * y), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000002e141

   1. Initial program 95.5%

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

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if 1.00000000000000002e141 < (*.f64 z z)

   1. Initial program 71.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 85.3%

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

Alternative 8: 58.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* x x) 6.2e+71) (* (* 4.0 y) t) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 6.2e+71) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    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) <= 6.2d+71) then
        tmp = (4.0d0 * y) * t
    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) <= 6.2e+71) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (x * x) <= 6.2e+71:
		tmp = (4.0 * y) * t
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * x) <= 6.2e+71)
		tmp = Float64(Float64(4.0 * y) * t);
	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) <= 6.2e+71)
		tmp = (4.0 * y) * t;
	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[(x * x), $MachinePrecision], 6.2e+71], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.20000000000000036e71

   1. Initial program 91.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

   if 6.20000000000000036e71 < (*.f64 x x)

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 9: 40.7% 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 = abs(z)
real(8) function code(x, y, z_m, t)
    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 86.3%

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

3. Taylor expanded in y around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 42.5

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

5. Applied rewrites 42.5%

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

Developer Target 1: 90.8% 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 2024276 
(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))))