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

Percentage Accurate: 90.7% → 96.3%

Time: 11.8s

Alternatives: 8

Speedup: 0.6×

• 49.0% 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: 90.7% 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: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+282)
   (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) <= 1e+282) {
		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) <= 1e+282)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+282], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000003e282

   1. Initial program 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. distribute-lft-neg-out 98.9%

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

      3. +-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. associate-*l* 98.9%

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

      7. distribute-rgt-neg-in 98.9%

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

      8. fma-define 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. remove-double-neg 99.4%

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

      13. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   if 1.00000000000000003e282 < (*.f64 z z)

   1. Initial program 71.4%

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

      1. fma-neg 79.2%

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

      2. distribute-lft-neg-in 79.2%

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

      3. *-commutative 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. metadata-eval 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in z around inf 79.2%

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

      1. associate-*r* 79.2%

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

      2. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. add-cube-cbrt 79.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}} \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right) \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}} \]

      2. pow3 79.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right)}^{3}} \]

      3. associate-*l* 79.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(-4 \cdot {z}^{2}\right)}}\right)}^{3} \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(-4 \cdot {z}^{2}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 79.2%

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

       2. associate-*r* 79.2%

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

       3. metadata-eval 79.2%

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

       4. distribute-rgt-neg-in 79.2%

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

       5. unpow2 79.2%

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

       6. associate-*r* 92.1%

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

       7. distribute-rgt-neg-in 92.1%

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

       8. metadata-eval 92.1%

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

   11. Applied egg-rr 92.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+282)
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+282) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+282)
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+282], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000003e282

   1. Initial program 98.9%

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

      1. fma-neg 99.0%

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

      2. distribute-lft-neg-in 99.0%

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

      3. *-commutative 99.0%

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

      4. distribute-rgt-neg-in 99.0%

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

      5. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   if 1.00000000000000003e282 < (*.f64 z z)

   1. Initial program 71.4%

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

      1. fma-neg 79.2%

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

      2. distribute-lft-neg-in 79.2%

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

      3. *-commutative 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. metadata-eval 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in z around inf 79.2%

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

      1. associate-*r* 79.2%

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

      2. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. add-cube-cbrt 79.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}} \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right) \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}} \]

      2. pow3 79.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right)}^{3}} \]

      3. associate-*l* 79.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(-4 \cdot {z}^{2}\right)}}\right)}^{3} \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(-4 \cdot {z}^{2}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 79.2%

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

       2. associate-*r* 79.2%

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

       3. metadata-eval 79.2%

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

       4. distribute-rgt-neg-in 79.2%

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

       5. unpow2 79.2%

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

       6. associate-*r* 92.1%

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

       7. distribute-rgt-neg-in 92.1%

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

       8. metadata-eval 92.1%

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

   11. Applied egg-rr 92.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+282)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+282) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} 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+282) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    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+282) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+282:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+282)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+282)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	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+282], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000003e282

   1. Initial program 98.9%

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

   if 1.00000000000000003e282 < (*.f64 z z)

   1. Initial program 71.4%

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

      1. fma-neg 79.2%

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

      2. distribute-lft-neg-in 79.2%

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

      3. *-commutative 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. metadata-eval 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in z around inf 79.2%

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

      1. associate-*r* 79.2%

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

      2. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. add-cube-cbrt 79.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}} \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right) \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}} \]

      2. pow3 79.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right)}^{3}} \]

      3. associate-*l* 79.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(-4 \cdot {z}^{2}\right)}}\right)}^{3} \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(-4 \cdot {z}^{2}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 79.2%

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

       2. associate-*r* 79.2%

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

       3. metadata-eval 79.2%

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

       4. distribute-rgt-neg-in 79.2%

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

       5. unpow2 79.2%

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

       6. associate-*r* 92.1%

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

       7. distribute-rgt-neg-in 92.1%

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

       8. metadata-eval 92.1%

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

   11. Applied egg-rr 92.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+282}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.3 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right) + 4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.3e+74)
   (+ (* y (* (* z z) -4.0)) (* 4.0 (* y t)))
   (- (* x x) (* y (* t -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.3e+74) {
		tmp = (y * ((z * z) * -4.0)) + (4.0 * (y * t));
	} else {
		tmp = (x * x) - (y * (t * -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) <= 1.3d+74) then
        tmp = (y * ((z * z) * (-4.0d0))) + (4.0d0 * (y * t))
    else
        tmp = (x * x) - (y * (t * (-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) <= 1.3e+74) {
		tmp = (y * ((z * z) * -4.0)) + (4.0 * (y * t));
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.3e+74:
		tmp = (y * ((z * z) * -4.0)) + (4.0 * (y * t))
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.3e+74)
		tmp = Float64(Float64(y * Float64(Float64(z * z) * -4.0)) + Float64(4.0 * Float64(y * t)));
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.3e+74)
		tmp = (y * ((z * z) * -4.0)) + (4.0 * (y * t));
	else
		tmp = (x * x) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.3e+74], N[(N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.3e74

   1. Initial program 96.3%

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

   3. Taylor expanded in z around 0 96.3%

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

   4. Taylor expanded in x around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. distribute-lft-in 87.9%

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

      3. associate-*r* 87.3%

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

      4. *-commutative 87.3%

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

      5. unpow2 87.3%

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

      6. add-sqr-sqrt 36.9%

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

      7. associate-*l* 37.0%

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

      8. associate-*r* 37.0%

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

      9. neg-mul-1 37.0%

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

      10. distribute-rgt-neg-in 37.0%

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

      11. metadata-eval 37.0%

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

      12. associate-*r* 37.6%

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

      13. *-commutative 37.6%

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

      14. associate-*l* 37.6%

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

      15. add-sqr-sqrt 87.9%

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

      16. unpow2 87.9%

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

      17. associate-*r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. unpow2 87.9%

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

   8. Applied egg-rr 87.9%

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

   if 1.3e74 < (*.f64 x x)

   1. Initial program 88.7%

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

   3. Taylor expanded in z around 0 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

      3. associate-*l* 83.1%

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

   5. Simplified 83.1%

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

Alternative 5: 46.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.15e-82)
   (* 4.0 (* y t))
   (if (<= x 1.2e+37) (* z (* z (* y -4.0))) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.15e-82) {
		tmp = 4.0 * (y * t);
	} else if (x <= 1.2e+37) {
		tmp = z * (z * (y * -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 <= 1.15d-82) then
        tmp = 4.0d0 * (y * t)
    else if (x <= 1.2d+37) then
        tmp = z * (z * (y * (-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 <= 1.15e-82) {
		tmp = 4.0 * (y * t);
	} else if (x <= 1.2e+37) {
		tmp = z * (z * (y * -4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.15e-82:
		tmp = 4.0 * (y * t)
	elif x <= 1.2e+37:
		tmp = z * (z * (y * -4.0))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.15e-82)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (x <= 1.2e+37)
		tmp = Float64(z * Float64(z * Float64(y * -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 <= 1.15e-82)
		tmp = 4.0 * (y * t);
	elseif (x <= 1.2e+37)
		tmp = z * (z * (y * -4.0));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.15e-82], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+37], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.14999999999999998e-82

   1. Initial program 95.0%

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

      1. fma-neg 96.7%

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

      2. distribute-lft-neg-in 96.7%

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

      3. *-commutative 96.7%

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

      4. distribute-rgt-neg-in 96.7%

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

      5. metadata-eval 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in t around inf 45.6%

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

      1. *-commutative 45.6%

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

   7. Simplified 45.6%

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

   if 1.14999999999999998e-82 < x < 1.2e37

   1. Initial program 92.4%

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

      1. fma-neg 92.4%

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

      2. distribute-lft-neg-in 92.4%

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

      3. *-commutative 92.4%

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

      4. distribute-rgt-neg-in 92.4%

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

      5. metadata-eval 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in z around inf 53.0%

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

      1. associate-*r* 53.0%

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

      2. *-commutative 53.0%

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

   7. Simplified 53.0%

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

      1. add-cube-cbrt 52.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}} \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right) \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}} \]

      2. pow3 52.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right)}^{3}} \]

      3. associate-*l* 52.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(-4 \cdot {z}^{2}\right)}}\right)}^{3} \]

   9. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(-4 \cdot {z}^{2}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 53.0%

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

       2. associate-*r* 53.0%

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

       3. metadata-eval 53.0%

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

       4. distribute-rgt-neg-in 53.0%

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

       5. unpow2 53.0%

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

       6. associate-*r* 60.3%

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

       7. distribute-rgt-neg-in 60.3%

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

       8. metadata-eval 60.3%

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

   11. Applied egg-rr 60.3%

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

   if 1.2e37 < x

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0 88.8%

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

   5. Simplified 74.7%

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

      1. --rgt-identity 74.7%

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

   7. Applied egg-rr 74.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+64}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1e+64) (- (* 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 <= 1e+64) {
		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 <= 1d+64) 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 <= 1e+64) {
		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 <= 1e+64:
		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 (z <= 1e+64)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1e+64)
		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[z, 1e+64], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000002e64

   1. Initial program 95.3%

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

   3. Taylor expanded in z around 0 73.7%

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

      1. *-commutative 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*l* 73.2%

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

   5. Simplified 73.2%

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

   if 1.00000000000000002e64 < z

   1. Initial program 84.7%

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

      1. fma-neg 86.9%

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

      2. distribute-lft-neg-in 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt-neg-in 86.9%

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

      5. metadata-eval 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in z around inf 74.6%

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

      1. associate-*r* 74.6%

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

      2. *-commutative 74.6%

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

   7. Simplified 74.6%

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

      1. add-cube-cbrt 74.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}} \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right) \cdot \sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}} \]

      2. pow3 74.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(y \cdot -4\right) \cdot {z}^{2}}\right)}^{3}} \]

      3. associate-*l* 74.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(-4 \cdot {z}^{2}\right)}}\right)}^{3} \]

   9. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(-4 \cdot {z}^{2}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 74.6%

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

       2. associate-*r* 74.6%

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

       3. metadata-eval 74.6%

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

       4. distribute-rgt-neg-in 74.6%

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

       5. unpow2 74.6%

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

       6. associate-*r* 78.7%

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

       7. distribute-rgt-neg-in 78.7%

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

       8. metadata-eval 78.7%

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

   11. Applied egg-rr 78.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+64}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+17}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.8e+17) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.8e+17) {
		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 <= 7.8d+17) 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 <= 7.8e+17) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 7.8e+17:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.8e+17)
		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 <= 7.8e+17)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.8e+17], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.8e17

   1. Initial program 94.5%

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

      1. fma-neg 96.1%

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

      2. distribute-lft-neg-in 96.1%

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

      3. *-commutative 96.1%

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

      4. distribute-rgt-neg-in 96.1%

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

      5. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in t around inf 43.2%

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

      1. *-commutative 43.2%

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

   7. Simplified 43.2%

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

   if 7.8e17 < x

   1. Initial program 89.9%

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

   3. Taylor expanded in y around 0 89.9%

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

   5. Simplified 69.0%

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

      1. --rgt-identity 69.0%

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

   7. Applied egg-rr 69.0%

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

Alternative 8: 40.8% 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 93.5%

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

3. Taylor expanded in y around 0 93.8%

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

5. Simplified 35.8%

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

   1. --rgt-identity 35.8%

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

7. Applied egg-rr 35.8%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024145 
(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))))