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

Percentage Accurate: 90.0% → 95.7%

Time: 9.2s

Alternatives: 7

Speedup: 0.8×

• 48.9% 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: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+267)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* x x) (* z (* y (* z 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+267) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - (z * (y * (z * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+267)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(y * Float64(z * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+267], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e267

   1. Initial program 98.8%

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

      1. fma-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   if 4.9999999999999999e267 < (*.f64 z z)

   1. Initial program 69.5%

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

      1. flip-- 1.5%

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

      2. associate-*r/ 1.5%

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

      3. pow2 1.5%

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

      4. metadata-eval 1.5%

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

      5. pow-prod-down 1.5%

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

      6. pow-prod-up 1.5%

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

      7. metadata-eval 1.5%

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

      8. metadata-eval 1.5%

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

      9. metadata-eval 1.5%

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

      10. fma-def 1.5%

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

   3. Applied egg-rr 1.5%

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

      1. associate-/l* 1.5%

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

      2. *-commutative 1.5%

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

   5. Simplified 1.5%

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

   6. Taylor expanded in z around inf 69.5%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{z \cdot z}}} \]

   8. Simplified 69.5%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{z \cdot z}}} \]
   9. Step-by-step derivation

      1. pow2 69.5%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{{z}^{2}}}} \]

      2. metadata-eval 69.5%

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

      3. pow-flip 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{{z}^{\color{blue}{-2}}} \]

   10. Applied egg-rr 71.2%

       \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{{z}^{-2}}} \]

   11. Taylor expanded in y around 0 69.5%

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

       1. unpow2 69.5%

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

       2. associate-*r* 69.5%

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

       3. associate-*r* 93.4%

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

       4. *-commutative 93.4%

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

       5. associate-*r* 93.4%

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

   13. Simplified 93.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \]

Alternative 2: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.3 \cdot 10^{-308}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{-50}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1.3e-308)
   (* x x)
   (if (<= (* z z) 2e-50)
     (* 4.0 (* t y))
     (if (<= (* z z) 2.55e+57) (* x x) (* y (* (* z z) -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.3e-308) {
		tmp = x * x;
	} else if ((z * z) <= 2e-50) {
		tmp = 4.0 * (t * y);
	} else if ((z * z) <= 2.55e+57) {
		tmp = x * x;
	} else {
		tmp = y * ((z * 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) <= 1.3d-308) then
        tmp = x * x
    else if ((z * z) <= 2d-50) then
        tmp = 4.0d0 * (t * y)
    else if ((z * z) <= 2.55d+57) then
        tmp = x * x
    else
        tmp = y * ((z * 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) <= 1.3e-308) {
		tmp = x * x;
	} else if ((z * z) <= 2e-50) {
		tmp = 4.0 * (t * y);
	} else if ((z * z) <= 2.55e+57) {
		tmp = x * x;
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1.3e-308:
		tmp = x * x
	elif (z * z) <= 2e-50:
		tmp = 4.0 * (t * y)
	elif (z * z) <= 2.55e+57:
		tmp = x * x
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1.3e-308)
		tmp = Float64(x * x);
	elseif (Float64(z * z) <= 2e-50)
		tmp = Float64(4.0 * Float64(t * y));
	elseif (Float64(z * z) <= 2.55e+57)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1.3e-308)
		tmp = x * x;
	elseif ((z * z) <= 2e-50)
		tmp = 4.0 * (t * y);
	elseif ((z * z) <= 2.55e+57)
		tmp = x * x;
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.3e-308], N[(x * x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e-50], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2.55e+57], N[(x * x), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.3e-308 or 2.00000000000000002e-50 < (*.f64 z z) < 2.55000000000000011e57

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 64.2%

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

      1. unpow2 64.2%

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

   4. Simplified 64.2%

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

   if 1.3e-308 < (*.f64 z z) < 2.00000000000000002e-50

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 60.2%

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

      1. *-commutative 60.2%

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

   4. Simplified 60.2%

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

   if 2.55000000000000011e57 < (*.f64 z z)

   1. Initial program 79.7%

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

   2. Taylor expanded in z around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. unpow2 70.1%

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

      3. associate-*l* 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.3 \cdot 10^{-308}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{-50}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]

Alternative 3: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+267}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+267)
   (+ (* x x) (* (* 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) <= 5e+267) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (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) <= 5d+267) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (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) <= 5e+267) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (y * (z * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+267:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (y * (z * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+267)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - 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) <= 5e+267)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (y * (z * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+267], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e267

   1. Initial program 98.8%

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

   if 4.9999999999999999e267 < (*.f64 z z)

   1. Initial program 69.5%

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

      1. flip-- 1.5%

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

      2. associate-*r/ 1.5%

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

      3. pow2 1.5%

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

      4. metadata-eval 1.5%

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

      5. pow-prod-down 1.5%

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

      6. pow-prod-up 1.5%

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

      7. metadata-eval 1.5%

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

      8. metadata-eval 1.5%

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

      9. metadata-eval 1.5%

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

      10. fma-def 1.5%

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

   3. Applied egg-rr 1.5%

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

      1. associate-/l* 1.5%

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

      2. *-commutative 1.5%

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

   5. Simplified 1.5%

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

   6. Taylor expanded in z around inf 69.5%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{z \cdot z}}} \]

   8. Simplified 69.5%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{z \cdot z}}} \]
   9. Step-by-step derivation

      1. pow2 69.5%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{{z}^{2}}}} \]

      2. metadata-eval 69.5%

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

      3. pow-flip 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{{z}^{\color{blue}{-2}}} \]

   10. Applied egg-rr 71.2%

       \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{{z}^{-2}}} \]

   11. Taylor expanded in y around 0 69.5%

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

       1. unpow2 69.5%

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

       2. associate-*r* 69.5%

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

       3. associate-*r* 93.4%

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

       4. *-commutative 93.4%

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

       5. associate-*r* 93.4%

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

   13. Simplified 93.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+267}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e-24)
   (- (* x x) (* y (* t -4.0)))
   (- (* x x) (* z (* y (* z 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e-24) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - (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) <= 2d-24) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (x * x) - (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) <= 2e-24) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (x * x) - (z * (y * (z * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e-24:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (x * x) - (z * (y * (z * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e-24)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(x * x) - 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) <= 2e-24)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (x * x) - (z * (y * (z * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-24], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999985e-24

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 97.6%

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

      1. associate-*r* 97.6%

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

   4. Simplified 97.6%

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

   if 1.99999999999999985e-24 < (*.f64 z z)

   1. Initial program 81.0%

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

      1. flip-- 27.1%

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

      2. associate-*r/ 25.1%

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

      3. pow2 25.1%

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

      4. metadata-eval 25.1%

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

      5. pow-prod-down 25.1%

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

      6. pow-prod-up 25.1%

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

      7. metadata-eval 25.1%

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

      8. metadata-eval 25.1%

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

      9. metadata-eval 25.1%

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

      10. fma-def 25.1%

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

   3. Applied egg-rr 25.1%

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

      1. associate-/l* 27.1%

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

      2. *-commutative 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in z around inf 76.2%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 76.2%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{z \cdot z}}} \]

   8. Simplified 76.2%

      \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{\frac{1}{z \cdot z}}} \]
   9. Step-by-step derivation

      1. pow2 76.2%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{\frac{1}{\color{blue}{{z}^{2}}}} \]

      2. metadata-eval 76.2%

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

      3. pow-flip 77.2%

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

      4. metadata-eval 77.2%

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

      5. metadata-eval 77.2%

         \[\leadsto x \cdot x - \frac{4 \cdot y}{{z}^{\color{blue}{-2}}} \]

   10. Applied egg-rr 77.2%

       \[\leadsto x \cdot x - \frac{4 \cdot y}{\color{blue}{{z}^{-2}}} \]

   11. Taylor expanded in y around 0 76.2%

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

       1. unpow2 76.2%

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

       2. associate-*r* 76.2%

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

       3. associate-*r* 89.9%

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

       4. *-commutative 89.9%

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

       5. associate-*r* 89.9%

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

   13. Simplified 89.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot \left(z \cdot 4\right)\right)\\ \end{array} \]

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.55 \cdot 10^{+76}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2.55e+76)
   (- (* x x) (* y (* t -4.0)))
   (* y (* (* z z) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2.55e+76) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * 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) <= 2.55d+76) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = y * ((z * 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) <= 2.55e+76) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2.55e+76:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2.55e+76)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2.55e+76)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2.55e+76], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.5500000000000001e76

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 93.6%

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

      1. associate-*r* 93.6%

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

   4. Simplified 93.6%

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

   if 2.5500000000000001e76 < (*.f64 z z)

   1. Initial program 78.5%

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

   2. Taylor expanded in z around inf 71.5%

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

      1. *-commutative 71.5%

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

      2. unpow2 71.5%

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

      3. associate-*l* 71.5%

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

   4. Simplified 71.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.55 \cdot 10^{+76}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]

Alternative 6: 44.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+37}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 4.5e+37) (* 4.0 (* t y)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.5e+37) {
		tmp = 4.0 * (t * y);
	} 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 <= 4.5d+37) then
        tmp = 4.0d0 * (t * y)
    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 <= 4.5e+37) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 4.5e+37:
		tmp = 4.0 * (t * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4.5e+37)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 4.5e+37)
		tmp = 4.0 * (t * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 4.5e+37], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.49999999999999962e37

   1. Initial program 90.4%

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

   2. Taylor expanded in t around inf 34.1%

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

      1. *-commutative 34.1%

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

   4. Simplified 34.1%

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

   if 4.49999999999999962e37 < x

   1. Initial program 88.2%

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

   2. Taylor expanded in x around inf 74.5%

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

      1. unpow2 74.5%

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

   4. Simplified 74.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+37}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 41.0% 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 89.9%

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

2. Taylor expanded in x around inf 39.5%

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

   1. unpow2 39.5%

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

4. Simplified 39.5%

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

5. Final simplification 39.5%

   \[\leadsto x \cdot x \]

Developer target: 90.1% 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 2023297 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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