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

Percentage Accurate: 90.7% → 95.9%

Time: 12.6s

Alternatives: 9

Speedup: 0.6×

• 49.5% 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: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+264)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* (* z (* z y)) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+264) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+264)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+264], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000009e264

   1. Initial program 95.5%

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

      1. cancel-sign-sub-inv 95.5%

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

      2. distribute-lft-neg-out 95.5%

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

      3. +-commutative 95.5%

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

      4. distribute-lft-neg-out 95.5%

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

      5. distribute-lft-neg-in 95.5%

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

      6. distribute-rgt-neg-in 95.5%

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

      7. fma-define 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. remove-double-neg 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   if 2.00000000000000009e264 < (*.f64 z z)

   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 0 67.2%

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

   4. Taylor expanded in z around inf 75.0%

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

      1. *-commutative 75.0%

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

   6. Simplified 75.0%

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

      1. add-sqr-sqrt 38.1%

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

      2. pow2 38.1%

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

      3. *-commutative 38.1%

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

      4. sqrt-prod 38.1%

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

      5. sqrt-pow1 44.1%

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

      6. metadata-eval 44.1%

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

      7. pow1 44.1%

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

   8. Applied egg-rr 44.1%

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

      1. unpow2 44.1%

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

      2. swap-sqr 38.1%

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

      3. add-sqr-sqrt 75.0%

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

      4. *-commutative 75.0%

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

      5. associate-*r* 84.7%

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

   10. Applied egg-rr 84.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+305}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{{x}^{2}}{t} - y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e+305)
   (+ (* x x) (* t (* 4.0 (- y (/ (* z y) (/ t z))))))
   (* t (- (/ (pow x 2.0) t) (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e+305) {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	} else {
		tmp = t * ((pow(x, 2.0) / t) - (y * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1d+305) then
        tmp = (x * x) + (t * (4.0d0 * (y - ((z * y) / (t / z)))))
    else
        tmp = t * (((x ** 2.0d0) / t) - (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e+305) {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	} else {
		tmp = t * ((Math.pow(x, 2.0) / t) - (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1e+305:
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))))
	else:
		tmp = t * ((math.pow(x, 2.0) / t) - (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1e+305)
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y - Float64(Float64(z * y) / Float64(t / z))))));
	else
		tmp = Float64(t * Float64(Float64((x ^ 2.0) / t) - Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1e+305)
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	else
		tmp = t * (((x ^ 2.0) / t) - (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+305], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y - N[(N[(z * y), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(N[Power[x, 2.0], $MachinePrecision] / t), $MachinePrecision] - N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999994e304

   1. Initial program 93.8%

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

   3. Taylor expanded in t around inf 90.6%

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

      1. +-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. metadata-eval 90.6%

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

      5. distribute-rgt-neg-in 90.6%

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

      6. distribute-lft-neg-in 90.6%

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

      7. distribute-rgt-out 90.6%

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

      8. unsub-neg 90.6%

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

      9. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

      1. unpow2 89.0%

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

      2. *-un-lft-identity 89.0%

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

      3. times-frac 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. /-rgt-identity 90.1%

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

      2. associate-*r* 94.2%

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

      3. clear-num 94.2%

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

      4. un-div-inv 94.2%

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

   9. Applied egg-rr 94.2%

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

   if 9.9999999999999994e304 < (*.f64 x x)

   1. Initial program 72.6%

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

   3. Taylor expanded in z around 0 84.9%

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

      1. *-commutative 84.9%

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

      2. *-commutative 84.9%

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

      3. associate-*l* 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in t around inf 94.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+305}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{{x}^{2}}{t} - y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-222}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{-58}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+147}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-222)
   (* x x)
   (if (<= (* z z) 2e-58)
     (* 4.0 (* y t))
     (if (<= (* z z) 1e+147) (* x x) (* (* z (* z y)) -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-222) {
		tmp = x * x;
	} else if ((z * z) <= 2e-58) {
		tmp = 4.0 * (y * t);
	} else if ((z * z) <= 1e+147) {
		tmp = x * x;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 5d-222) then
        tmp = x * x
    else if ((z * z) <= 2d-58) then
        tmp = 4.0d0 * (y * t)
    else if ((z * z) <= 1d+147) then
        tmp = x * x
    else
        tmp = (z * (z * y)) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-222) {
		tmp = x * x;
	} else if ((z * z) <= 2e-58) {
		tmp = 4.0 * (y * t);
	} else if ((z * z) <= 1e+147) {
		tmp = x * x;
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e-222:
		tmp = x * x
	elif (z * z) <= 2e-58:
		tmp = 4.0 * (y * t)
	elif (z * z) <= 1e+147:
		tmp = x * x
	else:
		tmp = (z * (z * y)) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-222)
		tmp = Float64(x * x);
	elseif (Float64(z * z) <= 2e-58)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (Float64(z * z) <= 1e+147)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e-222)
		tmp = x * x;
	elseif ((z * z) <= 2e-58)
		tmp = 4.0 * (y * t);
	elseif ((z * z) <= 1e+147)
		tmp = x * x;
	else
		tmp = (z * (z * y)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-222], N[(x * x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e-58], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+147], N[(x * x), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000008e-222 or 2.0000000000000001e-58 < (*.f64 z z) < 9.9999999999999998e146

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0 97.1%

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

   5. Simplified 63.5%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 63.5%

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

   7. Applied egg-rr 63.5%

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

   if 5.00000000000000008e-222 < (*.f64 z z) < 2.0000000000000001e-58

   1. Initial program 92.7%

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

      1. fmm-def 92.7%

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

      2. distribute-lft-neg-in 92.7%

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

      3. *-commutative 92.7%

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

      4. distribute-rgt-neg-in 92.7%

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

      5. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in t around inf 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   if 9.9999999999999998e146 < (*.f64 z z)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 71.1%

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

   4. Taylor expanded in z around inf 70.4%

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

      1. *-commutative 70.4%

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

   6. Simplified 70.4%

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

      1. add-sqr-sqrt 35.5%

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

      2. pow2 35.5%

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

      3. *-commutative 35.5%

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

      4. sqrt-prod 35.5%

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

      5. sqrt-pow1 39.8%

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

      6. metadata-eval 39.8%

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

      7. pow1 39.8%

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

   8. Applied egg-rr 39.8%

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

      1. unpow2 39.8%

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

      2. swap-sqr 35.6%

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

      3. add-sqr-sqrt 70.4%

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

      4. *-commutative 70.4%

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

      5. associate-*r* 77.2%

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

   10. Applied egg-rr 77.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-222}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{-58}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+147}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-221}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 1.8 \cdot 10^{-58}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e-221)
   (* x x)
   (if (<= (* z z) 1.8e-58)
     (* 4.0 (* y t))
     (if (<= (* z z) 1.45e+147) (* x x) (* -4.0 (* (* z z) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e-221) {
		tmp = x * x;
	} else if ((z * z) <= 1.8e-58) {
		tmp = 4.0 * (y * t);
	} else if ((z * z) <= 1.45e+147) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4d-221) then
        tmp = x * x
    else if ((z * z) <= 1.8d-58) then
        tmp = 4.0d0 * (y * t)
    else if ((z * z) <= 1.45d+147) then
        tmp = x * x
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e-221) {
		tmp = x * x;
	} else if ((z * z) <= 1.8e-58) {
		tmp = 4.0 * (y * t);
	} else if ((z * z) <= 1.45e+147) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e-221:
		tmp = x * x
	elif (z * z) <= 1.8e-58:
		tmp = 4.0 * (y * t)
	elif (z * z) <= 1.45e+147:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e-221)
		tmp = Float64(x * x);
	elseif (Float64(z * z) <= 1.8e-58)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (Float64(z * z) <= 1.45e+147)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e-221)
		tmp = x * x;
	elseif ((z * z) <= 1.8e-58)
		tmp = 4.0 * (y * t);
	elseif ((z * z) <= 1.45e+147)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e-221], N[(x * x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1.8e-58], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1.45e+147], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.00000000000000007e-221 or 1.80000000000000005e-58 < (*.f64 z z) < 1.4499999999999999e147

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0 97.1%

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

   5. Simplified 63.5%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 63.5%

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

   7. Applied egg-rr 63.5%

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

   if 4.00000000000000007e-221 < (*.f64 z z) < 1.80000000000000005e-58

   1. Initial program 92.7%

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

      1. fmm-def 92.7%

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

      2. distribute-lft-neg-in 92.7%

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

      3. *-commutative 92.7%

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

      4. distribute-rgt-neg-in 92.7%

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

      5. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in t around inf 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   if 1.4499999999999999e147 < (*.f64 z z)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 71.1%

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

   4. Taylor expanded in z around inf 70.4%

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

      1. *-commutative 70.4%

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

   6. Simplified 70.4%

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

      1. unpow2 70.4%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-221}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 1.8 \cdot 10^{-58}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+305)
   (+ (* x x) (* t (* 4.0 (- y (/ (* z y) (/ t z))))))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+305) {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 5d+305) then
        tmp = (x * x) + (t * (4.0d0 * (y - ((z * y) / (t / z)))))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+305) {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5e+305:
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+305)
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y - Float64(Float64(z * y) / Float64(t / z))))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5e+305)
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+305], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y - N[(N[(z * y), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000009e305

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. *-commutative 90.7%

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

      4. metadata-eval 90.7%

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

      5. distribute-rgt-neg-in 90.7%

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

      6. distribute-lft-neg-in 90.7%

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

      7. distribute-rgt-out 90.7%

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

      8. unsub-neg 90.7%

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

      9. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

      1. unpow2 89.1%

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

      2. *-un-lft-identity 89.1%

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

      3. times-frac 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. /-rgt-identity 90.1%

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

      2. associate-*r* 94.3%

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

      3. clear-num 94.3%

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

      4. un-div-inv 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 5.00000000000000009e305 < (*.f64 x x)

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0 72.2%

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

   5. Simplified 94.4%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 94.4%

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

   7. Applied egg-rr 94.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq \infty:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) INFINITY) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

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

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= math.inf:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < +inf.0

   1. Initial program 87.8%

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

   if +inf.0 < (*.f64 x x)

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0 87.8%

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

   5. Simplified 43.4%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 43.4%

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

   7. Applied egg-rr 43.4%

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

4. Final simplification 87.8%

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

Alternative 7: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+147}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+147) (- (* x x) (* y (* t -4.0))) (* (* z (* z y)) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+147) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+147) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (z * (z * y)) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+147) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z * (z * y)) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+147:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (z * (z * y)) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+147)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(z * Float64(z * y)) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+147)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (z * (z * y)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+147], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999998e146

   1. Initial program 95.9%

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

   3. Taylor expanded in z around 0 88.1%

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

      1. *-commutative 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-*l* 88.1%

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

   5. Simplified 88.1%

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

   if 9.9999999999999998e146 < (*.f64 z z)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0 71.1%

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

   4. Taylor expanded in z around inf 70.4%

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

      1. *-commutative 70.4%

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

   6. Simplified 70.4%

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

      1. add-sqr-sqrt 35.5%

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

      2. pow2 35.5%

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

      3. *-commutative 35.5%

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

      4. sqrt-prod 35.5%

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

      5. sqrt-pow1 39.8%

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

      6. metadata-eval 39.8%

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

      7. pow1 39.8%

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

   8. Applied egg-rr 39.8%

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

      1. unpow2 39.8%

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

      2. swap-sqr 35.6%

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

      3. add-sqr-sqrt 70.4%

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

      4. *-commutative 70.4%

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

      5. associate-*r* 77.2%

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

   10. Applied egg-rr 77.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+147}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.5e+48) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.5e+48) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 3.5d+48) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.5e+48) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 3.5e+48:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.5e+48)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 3.5e+48)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 3.5e+48], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999997e48

   1. Initial program 88.8%

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

      1. fmm-def 90.3%

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

      2. distribute-lft-neg-in 90.3%

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

      3. *-commutative 90.3%

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

      4. distribute-rgt-neg-in 90.3%

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

      5. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t around inf 33.2%

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

      1. *-commutative 33.2%

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

   7. Simplified 33.2%

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

   if 3.4999999999999997e48 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in y around 0 83.0%

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

   5. Simplified 78.2%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 78.2%

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

   7. Applied egg-rr 78.2%

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

Alternative 9: 40.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0 87.8%

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

5. Simplified 43.4%

   \[\leadsto x \cdot x - \color{blue}{0} \]
6. Step-by-step derivation

   1. --rgt-identity 43.4%

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

7. Applied egg-rr 43.4%

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

Developer Target 1: 90.7% 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 2024165 
(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))))