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

Percentage Accurate: 90.8% → 96.1%

Time: 7.8s

Alternatives: 8

Speedup: 0.6×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+293)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e292

   1. Initial program 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. distribute-lft-neg-out 98.4%

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

      3. +-commutative 98.4%

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

      4. associate-*l* 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. associate-*l* 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

      8. fma-define 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. remove-double-neg 100.0%

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

      13. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*l* 78.4%

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

   5. Simplified 78.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow2 37.2%

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

      3. pow2 37.2%

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

      4. associate-*r* 37.2%

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

      5. sqrt-prod 37.2%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow2 42.3%

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

      2. swap-sqr 37.2%

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

      3. add-sqr-sqrt 78.4%

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

      4. associate-*r* 78.4%

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

      5. associate-*r* 78.4%

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

      6. associate-*r* 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+293)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e292

   1. Initial program 98.4%

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

      1. fma-neg 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

      5. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*l* 78.4%

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

   5. Simplified 78.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow2 37.2%

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

      3. pow2 37.2%

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

      4. associate-*r* 37.2%

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

      5. sqrt-prod 37.2%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow2 42.3%

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

      2. swap-sqr 37.2%

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

      3. add-sqr-sqrt 78.4%

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

      4. associate-*r* 78.4%

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

      5. associate-*r* 78.4%

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

      6. associate-*r* 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-244}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.7e-244)
   (* 4.0 (* y t))
   (if (<= (* x x) 2.8e+86) (* z (* y (* z -4.0))) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.7e-244) {
		tmp = 4.0 * (y * t);
	} else if ((x * x) <= 2.8e+86) {
		tmp = z * (y * (z * -4.0));
	} 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) <= 1.7d-244) then
        tmp = 4.0d0 * (y * t)
    else if ((x * x) <= 2.8d+86) then
        tmp = z * (y * (z * (-4.0d0)))
    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) <= 1.7e-244) {
		tmp = 4.0 * (y * t);
	} else if ((x * x) <= 2.8e+86) {
		tmp = z * (y * (z * -4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.7e-244:
		tmp = 4.0 * (y * t)
	elif (x * x) <= 2.8e+86:
		tmp = z * (y * (z * -4.0))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.7e-244)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (Float64(x * x) <= 2.8e+86)
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.7e-244)
		tmp = 4.0 * (y * t);
	elseif ((x * x) <= 2.8e+86)
		tmp = z * (y * (z * -4.0));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.7e-244], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2.8e+86], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.70000000000000004e-244

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 61.1%

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

      1. *-commutative 61.1%

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

   5. Simplified 61.1%

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

   if 1.70000000000000004e-244 < (*.f64 x x) < 2.80000000000000004e86

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf 49.2%

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

      1. associate-*r* 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*l* 49.2%

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

   5. Simplified 49.2%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. pow2 18.7%

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

      4. associate-*r* 18.7%

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

      5. sqrt-prod 18.3%

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

      6. sqrt-prod 12.1%

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

      7. add-sqr-sqrt 21.9%

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

   7. Applied egg-rr 21.9%

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

      1. unpow2 21.9%

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

      2. swap-sqr 18.3%

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

      3. add-sqr-sqrt 49.2%

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

      4. associate-*r* 49.2%

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

      5. associate-*r* 49.2%

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

      6. associate-*r* 52.7%

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

   9. Applied egg-rr 52.7%

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

   if 2.80000000000000004e86 < (*.f64 x x)

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0 85.7%

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

   5. Simplified 77.5%

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

      1. --rgt-identity 77.5%

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

   7. Applied egg-rr 77.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-244}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.4 \cdot 10^{-246}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 4.4e-246)
   (* 4.0 (* y t))
   (if (<= (* x x) 4.4e+86) (* y (* (* z z) -4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 4.4e-246) {
		tmp = 4.0 * (y * t);
	} else if ((x * x) <= 4.4e+86) {
		tmp = y * ((z * z) * -4.0);
	} 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) <= 4.4d-246) then
        tmp = 4.0d0 * (y * t)
    else if ((x * x) <= 4.4d+86) then
        tmp = y * ((z * z) * (-4.0d0))
    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) <= 4.4e-246) {
		tmp = 4.0 * (y * t);
	} else if ((x * x) <= 4.4e+86) {
		tmp = y * ((z * z) * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 4.4e-246:
		tmp = 4.0 * (y * t)
	elif (x * x) <= 4.4e+86:
		tmp = y * ((z * z) * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 4.4e-246)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (Float64(x * x) <= 4.4e+86)
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 4.4e-246)
		tmp = 4.0 * (y * t);
	elseif ((x * x) <= 4.4e+86)
		tmp = y * ((z * z) * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.4e-246], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4.4e+86], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.39999999999999996e-246

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 61.1%

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

      1. *-commutative 61.1%

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

   5. Simplified 61.1%

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

   if 4.39999999999999996e-246 < (*.f64 x x) < 4.40000000000000006e86

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf 49.2%

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

      1. associate-*r* 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*l* 49.2%

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

   5. Simplified 49.2%

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

      1. pow2 49.2%

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

   7. Applied egg-rr 49.2%

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

   if 4.40000000000000006e86 < (*.f64 x x)

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0 85.7%

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

   5. Simplified 77.5%

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

      1. --rgt-identity 77.5%

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

   7. Applied egg-rr 77.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.4 \cdot 10^{-246}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+293)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+293) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (y * (z * (-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+293) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+293:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+293)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e292

   1. Initial program 98.4%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*l* 78.4%

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

   5. Simplified 78.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow2 37.2%

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

      3. pow2 37.2%

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

      4. associate-*r* 37.2%

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

      5. sqrt-prod 37.2%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow2 42.3%

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

      2. swap-sqr 37.2%

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

      3. add-sqr-sqrt 78.4%

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

      4. associate-*r* 78.4%

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

      5. associate-*r* 78.4%

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

      6. associate-*r* 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+293) (- (* x x) (* y (* t -4.0))) (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+293) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = z * (y * (z * (-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+293) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+293:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+293)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e292

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0 89.5%

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

      1. *-commutative 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*l* 89.5%

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

   5. Simplified 89.5%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*l* 78.4%

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

   5. Simplified 78.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow2 37.2%

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

      3. pow2 37.2%

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

      4. associate-*r* 37.2%

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

      5. sqrt-prod 37.2%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow2 42.3%

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

      2. swap-sqr 37.2%

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

      3. add-sqr-sqrt 78.4%

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

      4. associate-*r* 78.4%

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

      5. associate-*r* 78.4%

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

      6. associate-*r* 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.05 \cdot 10^{+27}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.05e+27) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.05e+27) {
		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 * x) <= 2.05d+27) 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 * x) <= 2.05e+27) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2.05e+27:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2.05e+27)
		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 * x) <= 2.05e+27)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.05e+27], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0500000000000001e27

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf 51.6%

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

      1. *-commutative 51.6%

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

   5. Simplified 51.6%

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

   if 2.0500000000000001e27 < (*.f64 x x)

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0 86.6%

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

   5. Simplified 74.4%

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

      1. --rgt-identity 74.4%

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

   7. Applied egg-rr 74.4%

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

Alternative 8: 41.3% 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 90.5%

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

3. Taylor expanded in y around 0 90.5%

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

5. Simplified 44.0%

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

   1. --rgt-identity 44.0%

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

7. Applied egg-rr 44.0%

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

Developer Target 1: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(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))))