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

Percentage Accurate: 90.9% → 94.5%

Time: 12.0s

Alternatives: 6

Speedup: 0.5×

• 50.1% 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: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z\_m \cdot z\_m - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;z\_m \leq 7.5 \cdot 10^{+220}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - z\_m \cdot \frac{y \cdot \left(z\_m \cdot \left(t \cdot 4\right)\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z\_m \cdot \frac{z\_m}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 4.6e+154)
   (fma x x (* (- (* z_m z_m) t) (* y -4.0)))
   (if (<= z_m 7.5e+220)
     (+ (* x x) (- (* y (* t 4.0)) (* z_m (/ (* y (* z_m (* t 4.0))) t))))
     (* -4.0 (* t (* y (+ -1.0 (* z_m (/ z_m t)))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.6e+154) {
		tmp = fma(x, x, (((z_m * z_m) - t) * (y * -4.0)));
	} else if (z_m <= 7.5e+220) {
		tmp = (x * x) + ((y * (t * 4.0)) - (z_m * ((y * (z_m * (t * 4.0))) / t)));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.6e+154)
		tmp = fma(x, x, Float64(Float64(Float64(z_m * z_m) - t) * Float64(y * -4.0)));
	elseif (z_m <= 7.5e+220)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * Float64(t * 4.0)) - Float64(z_m * Float64(Float64(y * Float64(z_m * Float64(t * 4.0))) / t))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * Float64(-1.0 + Float64(z_m * Float64(z_m / t))))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 4.6e+154], N[(x * x + N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 7.5e+220], N[(N[(x * x), $MachinePrecision] + N[(N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * N[(N[(y * N[(z$95$m * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(-1.0 + N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.6e154

   1. Initial program 95.6%

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

      1. fmm-def 96.9%

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

      2. distribute-lft-neg-in 96.9%

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

      3. *-commutative 96.9%

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

      4. distribute-rgt-neg-in 96.9%

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

      5. metadata-eval 96.9%

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

   3. Simplified 96.9%

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

   if 4.6e154 < z < 7.5000000000000003e220

   1. Initial program 54.8%

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

   3. Taylor expanded in t around inf 54.8%

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

      1. unpow2 54.8%

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

      2. associate-/l* 54.8%

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

      3. fmm-def 54.8%

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

      4. metadata-eval 54.8%

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

   5. Simplified 54.8%

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

      1. associate-*r* 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*r* 60.8%

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

      4. fma-undefine 60.8%

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

      5. distribute-lft-in 60.8%

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

      6. associate-*r* 60.8%

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

      7. *-commutative 60.8%

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

      8. associate-*l* 60.8%

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

      9. associate-*r* 60.8%

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

      10. *-commutative 60.8%

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

      11. associate-*l* 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. associate-*r* 73.2%

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

      2. clear-num 73.2%

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

      3. un-div-inv 73.1%

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

   9. Applied egg-rr 73.1%

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

       1. associate-/r/ 86.8%

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

       2. associate-*l* 86.8%

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

   11. Applied egg-rr 86.8%

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

   if 7.5000000000000003e220 < z

   1. Initial program 75.1%

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

   3. Taylor expanded in t around inf 75.1%

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

      1. unpow2 75.1%

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

      2. associate-/l* 75.1%

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

      3. fmm-def 75.1%

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

      4. metadata-eval 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in x around 0 85.7%

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

      1. unpow2 85.7%

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

      2. associate-*r/ 90.4%

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

      3. *-commutative 90.4%

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

   8. Applied egg-rr 90.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+220}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - z \cdot \frac{y \cdot \left(z \cdot \left(t \cdot 4\right)\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.85 \cdot 10^{+132}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z\_m \cdot z\_m\right)\\ \mathbf{elif}\;z\_m \leq 5 \cdot 10^{+220}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - z\_m \cdot \frac{y \cdot \left(z\_m \cdot \left(t \cdot 4\right)\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z\_m \cdot \frac{z\_m}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.85e+132)
   (+ (* x x) (* (* y 4.0) (- t (* z_m z_m))))
   (if (<= z_m 5e+220)
     (+ (* x x) (- (* y (* t 4.0)) (* z_m (/ (* y (* z_m (* t 4.0))) t))))
     (* -4.0 (* t (* y (+ -1.0 (* z_m (/ z_m t)))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.85e+132) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} else if (z_m <= 5e+220) {
		tmp = (x * x) + ((y * (t * 4.0)) - (z_m * ((y * (z_m * (t * 4.0))) / t)));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	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 (z_m <= 1.85d+132) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z_m * z_m)))
    else if (z_m <= 5d+220) then
        tmp = (x * x) + ((y * (t * 4.0d0)) - (z_m * ((y * (z_m * (t * 4.0d0))) / t)))
    else
        tmp = (-4.0d0) * (t * (y * ((-1.0d0) + (z_m * (z_m / t)))))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.85e+132) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} else if (z_m <= 5e+220) {
		tmp = (x * x) + ((y * (t * 4.0)) - (z_m * ((y * (z_m * (t * 4.0))) / t)));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 1.85e+132:
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)))
	elif z_m <= 5e+220:
		tmp = (x * x) + ((y * (t * 4.0)) - (z_m * ((y * (z_m * (t * 4.0))) / t)))
	else:
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.85e+132)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z_m * z_m))));
	elseif (z_m <= 5e+220)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * Float64(t * 4.0)) - Float64(z_m * Float64(Float64(y * Float64(z_m * Float64(t * 4.0))) / t))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * Float64(-1.0 + Float64(z_m * Float64(z_m / t))))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 1.85e+132)
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	elseif (z_m <= 5e+220)
		tmp = (x * x) + ((y * (t * 4.0)) - (z_m * ((y * (z_m * (t * 4.0))) / t)));
	else
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 1.85e+132], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5e+220], N[(N[(x * x), $MachinePrecision] + N[(N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * N[(N[(y * N[(z$95$m * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(-1.0 + N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.85000000000000005e132

   1. Initial program 95.5%

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

   if 1.85000000000000005e132 < z < 5.0000000000000002e220

   1. Initial program 66.1%

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

   3. Taylor expanded in t around inf 61.1%

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

      1. unpow2 61.1%

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

      2. associate-/l* 61.1%

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

      3. fmm-def 61.1%

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

      4. metadata-eval 61.1%

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

   5. Simplified 61.1%

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

      1. associate-*r* 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*r* 65.6%

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

      4. fma-undefine 65.6%

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

      5. distribute-lft-in 65.6%

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

      6. associate-*r* 65.6%

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

      7. *-commutative 65.6%

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

      8. associate-*l* 65.6%

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

      9. associate-*r* 65.6%

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

      10. *-commutative 65.6%

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

      11. associate-*l* 65.6%

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

   7. Applied egg-rr 65.6%

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

      1. associate-*r* 74.9%

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

      2. clear-num 74.9%

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

      3. un-div-inv 74.8%

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

   9. Applied egg-rr 74.8%

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

       1. associate-/r/ 90.1%

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

       2. associate-*l* 90.1%

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

   11. Applied egg-rr 90.1%

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

   if 5.0000000000000002e220 < z

   1. Initial program 75.1%

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

   3. Taylor expanded in t around inf 75.1%

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

      1. unpow2 75.1%

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

      2. associate-/l* 75.1%

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

      3. fmm-def 75.1%

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

      4. metadata-eval 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in x around 0 85.7%

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

      1. unpow2 85.7%

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

      2. associate-*r/ 90.4%

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

      3. *-commutative 90.4%

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

   8. Applied egg-rr 90.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+132}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+220}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - z \cdot \frac{y \cdot \left(z \cdot \left(t \cdot 4\right)\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(z\_m \cdot z\_m - t\right) \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z\_m \cdot \frac{z\_m}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* (- (* z_m z_m) t) (* y 4.0)) 2e+295)
   (+ (* x x) (* (* y 4.0) (- t (* z_m z_m))))
   (* -4.0 (* t (* y (+ -1.0 (* z_m (/ z_m t))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((((z_m * z_m) - t) * (y * 4.0)) <= 2e+295) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	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 ((((z_m * z_m) - t) * (y * 4.0d0)) <= 2d+295) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z_m * z_m)))
    else
        tmp = (-4.0d0) * (t * (y * ((-1.0d0) + (z_m * (z_m / t)))))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((((z_m * z_m) - t) * (y * 4.0)) <= 2e+295) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (((z_m * z_m) - t) * (y * 4.0)) <= 2e+295:
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)))
	else:
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(Float64(z_m * z_m) - t) * Float64(y * 4.0)) <= 2e+295)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z_m * z_m))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * Float64(-1.0 + Float64(z_m * Float64(z_m / t))))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((((z_m * z_m) - t) * (y * 4.0)) <= 2e+295)
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	else
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 2e+295], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(-1.0 + N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 2e295

   1. Initial program 96.8%

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

   if 2e295 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 67.4%

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

   3. Taylor expanded in t around inf 67.4%

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

      1. unpow2 67.4%

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

      2. associate-/l* 67.4%

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

      3. fmm-def 67.4%

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

      4. metadata-eval 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in x around 0 78.5%

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

      1. unpow2 78.5%

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

      2. associate-*r/ 82.4%

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

      3. *-commutative 82.4%

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

   8. Applied egg-rr 82.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z\_m \cdot \frac{z\_m}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.2e+75)
   (- (* x x) (* y (* t -4.0)))
   (* -4.0 (* t (* y (+ -1.0 (* z_m (/ z_m t))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.2e+75) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	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 (z_m <= 1.2d+75) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (-4.0d0) * (t * (y * ((-1.0d0) + (z_m * (z_m / t)))))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.2e+75) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 1.2e+75:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.2e+75)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * Float64(-1.0 + Float64(z_m * Float64(z_m / t))))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 1.2e+75)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = -4.0 * (t * (y * (-1.0 + (z_m * (z_m / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 1.2e+75], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(-1.0 + N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e75

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*l* 74.2%

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

   5. Simplified 74.2%

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

   if 1.2e75 < z

   1. Initial program 78.3%

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

   3. Taylor expanded in t around inf 71.0%

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

      1. unpow2 71.0%

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

      2. associate-/l* 71.0%

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

      3. fmm-def 71.0%

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

      4. metadata-eval 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in x around 0 69.3%

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

      1. unpow2 69.3%

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

      2. associate-*r/ 72.7%

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

      3. *-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \left(-1 + z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) - (y * (t * (-4.0d0)))
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return (x * x) - (y * (t * -4.0));
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return (x * x) - (y * (t * -4.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in z around 0 66.5%

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

   1. *-commutative 66.5%

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

   2. *-commutative 66.5%

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

   3. associate-*l* 66.5%

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

5. Simplified 66.5%

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

6. Final simplification 66.5%

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

Alternative 6: 31.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 = 4.0d0 * (t * y)
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return 4.0 * (t * y);
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return 4.0 * (t * y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in t around inf 30.6%

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

   1. *-commutative 30.6%

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

5. Simplified 30.6%

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

6. Final simplification 30.6%

   \[\leadsto 4 \cdot \left(t \cdot y\right) \]
7. Add Preprocessing

Developer target: 90.9% 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 2024076 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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