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

Percentage Accurate: 90.7% → 95.6%

Time: 8.6s

Alternatives: 11

Speedup: 0.8×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{y \cdot -4}\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left({t_1}^{2}, t_1 \cdot \left(z \cdot z - t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (cbrt (* y -4.0))))
   (if (<= (* z z) 5e+246)
     (fma (pow t_1 2.0) (* t_1 (- (* z z) t)) (* x x))
     (- (* x x) (* z (* z (* y 4.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = cbrt((y * -4.0));
	double tmp;
	if ((z * z) <= 5e+246) {
		tmp = fma(pow(t_1, 2.0), (t_1 * ((z * z) - t)), (x * x));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = cbrt(Float64(y * -4.0))
	tmp = 0.0
	if (Float64(z * z) <= 5e+246)
		tmp = fma((t_1 ^ 2.0), Float64(t_1 * Float64(Float64(z * z) - t)), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Power[N[(y * -4.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e+246], N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999976e246

   1. Initial program 95.4%

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

      1. associate-*l* 96.0%

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

   3. Simplified 96.0%

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

      1. sub-neg 96.0%

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

      2. +-commutative 96.0%

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

      3. distribute-rgt-neg-in 96.0%

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

      4. distribute-lft-neg-in 96.0%

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

      5. metadata-eval 96.0%

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

      6. associate-*l* 95.4%

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

      7. add-cube-cbrt 94.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \sqrt[3]{y \cdot -4}\right)} \cdot \left(z \cdot z - t\right) + x \cdot x \]

      8. associate-*l* 94.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \left(\sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right)\right)} + x \cdot x \]

      9. fma-def 97.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)} \]

      10. pow2 97.7%

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

   5. Applied egg-rr 97.7%

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

   if 4.99999999999999976e246 < (*.f64 z z)

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around inf 79.9%

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

      1. *-commutative 79.9%

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

      2. unpow2 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*r* 79.9%

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

      5. associate-*l* 93.8%

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

      6. *-commutative 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{y \cdot -4}\right)}^{2}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 2: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+302)
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+302) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+302)
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+302], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e302

   1. Initial program 95.1%

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

      1. fma-neg 96.2%

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

      2. *-commutative 96.2%

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

      3. distribute-rgt-neg-in 96.2%

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

      4. distribute-rgt-neg-in 96.2%

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

      5. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

   if 2.0000000000000002e302 < (*.f64 z z)

   1. Initial program 78.5%

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

      1. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in z around inf 78.5%

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

      1. *-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-*r* 78.5%

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

      5. associate-*l* 94.3%

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

      6. *-commutative 94.3%

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

   6. Simplified 94.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 3: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x - \left(y \cdot -4\right) \cdot t\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+209}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x x) (* (* y -4.0) t))))
   (if (<= (* z z) 2e+140)
     t_1
     (if (<= (* z z) 2e+185)
       (* (* y -4.0) (- (* z z) t))
       (if (<= (* z z) 5e+209) t_1 (* z (* z (* y -4.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - ((y * -4.0) * t);
	double tmp;
	if ((z * z) <= 2e+140) {
		tmp = t_1;
	} else if ((z * z) <= 2e+185) {
		tmp = (y * -4.0) * ((z * z) - t);
	} else if ((z * z) <= 5e+209) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x * x) - ((y * (-4.0d0)) * t)
    if ((z * z) <= 2d+140) then
        tmp = t_1
    else if ((z * z) <= 2d+185) then
        tmp = (y * (-4.0d0)) * ((z * z) - t)
    else if ((z * z) <= 5d+209) then
        tmp = t_1
    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 t_1 = (x * x) - ((y * -4.0) * t);
	double tmp;
	if ((z * z) <= 2e+140) {
		tmp = t_1;
	} else if ((z * z) <= 2e+185) {
		tmp = (y * -4.0) * ((z * z) - t);
	} else if ((z * z) <= 5e+209) {
		tmp = t_1;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) - ((y * -4.0) * t)
	tmp = 0
	if (z * z) <= 2e+140:
		tmp = t_1
	elif (z * z) <= 2e+185:
		tmp = (y * -4.0) * ((z * z) - t)
	elif (z * z) <= 5e+209:
		tmp = t_1
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) - Float64(Float64(y * -4.0) * t))
	tmp = 0.0
	if (Float64(z * z) <= 2e+140)
		tmp = t_1;
	elseif (Float64(z * z) <= 2e+185)
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t));
	elseif (Float64(z * z) <= 5e+209)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) - ((y * -4.0) * t);
	tmp = 0.0;
	if ((z * z) <= 2e+140)
		tmp = t_1;
	elseif ((z * z) <= 2e+185)
		tmp = (y * -4.0) * ((z * z) - t);
	elseif ((z * z) <= 5e+209)
		tmp = t_1;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - N[(N[(y * -4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e+140], t$95$1, If[LessEqual[N[(z * z), $MachinePrecision], 2e+185], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+209], t$95$1, N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.00000000000000012e140 or 2e185 < (*.f64 z z) < 4.99999999999999964e209

   1. Initial program 95.5%

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

      1. associate-*l* 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in z around 0 89.9%

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

      1. associate-*r* 89.3%

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

      2. *-commutative 89.3%

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

   6. Simplified 89.3%

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

   if 2.00000000000000012e140 < (*.f64 z z) < 2e185

   1. Initial program 91.7%

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

      1. associate-*l* 91.7%

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

   3. Simplified 91.7%

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

      1. sub-neg 91.7%

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

      2. +-commutative 91.7%

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

      3. distribute-rgt-neg-in 91.7%

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

      4. distribute-lft-neg-in 91.7%

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

      5. metadata-eval 91.7%

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

      6. associate-*l* 91.7%

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

      7. add-cube-cbrt 90.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \sqrt[3]{y \cdot -4}\right)} \cdot \left(z \cdot z - t\right) + x \cdot x \]

      8. associate-*l* 90.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \left(\sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right)\right)} + x \cdot x \]

      9. fma-def 90.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)} \]

      10. pow2 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in x around 0 83.0%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot {1}^{0.3333333333333333}\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 83.0%

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

      2. *-rgt-identity 83.0%

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

      3. *-commutative 83.0%

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

      4. unpow2 83.0%

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

      5. distribute-rgt-out-- 83.0%

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

      6. distribute-lft-out-- 83.0%

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

      7. *-commutative 83.0%

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

      8. associate-*r* 83.0%

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

      9. *-commutative 83.0%

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

      10. *-commutative 83.0%

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

      11. associate-*r* 83.0%

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

      12. *-commutative 83.0%

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

      13. distribute-lft-out-- 83.0%

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

   8. Simplified 83.0%

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

   if 4.99999999999999964e209 < (*.f64 z z)

   1. Initial program 81.5%

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

      1. associate-*l* 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in z around inf 81.5%

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

      1. *-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-*r* 81.5%

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

      5. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+140}:\\ \;\;\;\;x \cdot x - \left(y \cdot -4\right) \cdot t\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+209}:\\ \;\;\;\;x \cdot x - \left(y \cdot -4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 57.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{if}\;x \cdot x \leq 1.2 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.45 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -4.0 (* (* z z) y))))
   (if (<= (* x x) 1.2e-307)
     t_1
     (if (<= (* x x) 3.4e-202)
       (* t (* y 4.0))
       (if (<= (* x x) 1.45e+126) t_1 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double tmp;
	if ((x * x) <= 1.2e-307) {
		tmp = t_1;
	} else if ((x * x) <= 3.4e-202) {
		tmp = t * (y * 4.0);
	} else if ((x * x) <= 1.45e+126) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * ((z * z) * y)
    if ((x * x) <= 1.2d-307) then
        tmp = t_1
    else if ((x * x) <= 3.4d-202) then
        tmp = t * (y * 4.0d0)
    else if ((x * x) <= 1.45d+126) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double tmp;
	if ((x * x) <= 1.2e-307) {
		tmp = t_1;
	} else if ((x * x) <= 3.4e-202) {
		tmp = t * (y * 4.0);
	} else if ((x * x) <= 1.45e+126) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -4.0 * ((z * z) * y)
	tmp = 0
	if (x * x) <= 1.2e-307:
		tmp = t_1
	elif (x * x) <= 3.4e-202:
		tmp = t * (y * 4.0)
	elif (x * x) <= 1.45e+126:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-4.0 * Float64(Float64(z * z) * y))
	tmp = 0.0
	if (Float64(x * x) <= 1.2e-307)
		tmp = t_1;
	elseif (Float64(x * x) <= 3.4e-202)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (Float64(x * x) <= 1.45e+126)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -4.0 * ((z * z) * y);
	tmp = 0.0;
	if ((x * x) <= 1.2e-307)
		tmp = t_1;
	elseif ((x * x) <= 3.4e-202)
		tmp = t * (y * 4.0);
	elseif ((x * x) <= 1.45e+126)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.2e-307], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 3.4e-202], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.45e+126], t$95$1, N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.20000000000000009e-307 or 3.40000000000000012e-202 < (*.f64 x x) < 1.44999999999999993e126

   1. Initial program 96.2%

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

      1. associate-*l* 96.2%

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

   3. Simplified 96.2%

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

      1. sub-neg 96.2%

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

      2. +-commutative 96.2%

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

      3. distribute-rgt-neg-in 96.2%

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

      4. distribute-lft-neg-in 96.2%

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

      5. metadata-eval 96.2%

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

      6. associate-*l* 96.2%

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

      7. add-cube-cbrt 95.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \sqrt[3]{y \cdot -4}\right)} \cdot \left(z \cdot z - t\right) + x \cdot x \]

      8. associate-*l* 95.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \left(\sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right)\right)} + x \cdot x \]

      9. fma-def 95.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)} \]

      10. pow2 95.5%

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

   5. Applied egg-rr 95.5%

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

   6. Taylor expanded in z around inf 55.6%

      \[\leadsto \color{blue}{-4 \cdot \left({1}^{0.3333333333333333} \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 55.6%

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

      2. unpow2 55.6%

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

      3. associate-*r* 55.6%

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

      4. *-lft-identity 55.6%

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

   8. Simplified 55.6%

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

   if 1.20000000000000009e-307 < (*.f64 x x) < 3.40000000000000012e-202

   1. Initial program 91.4%

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

      1. associate-*l* 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in t around inf 69.6%

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

      1. associate-*r* 69.6%

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

   6. Simplified 69.6%

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

   if 1.44999999999999993e126 < (*.f64 x x)

   1. Initial program 83.8%

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

      1. associate-*l* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in x around inf 79.5%

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

      1. unpow2 79.5%

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

   6. Simplified 79.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.2 \cdot 10^{-307}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.45 \cdot 10^{+126}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+61}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+140}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+40)
   (* x x)
   (if (<= (* z z) 2e+61)
     (* -4.0 (* (* z z) y))
     (if (<= (* z z) 2e+140) (* x x) (* z (* z (* y -4.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+40) {
		tmp = x * x;
	} else if ((z * z) <= 2e+61) {
		tmp = -4.0 * ((z * z) * y);
	} else if ((z * z) <= 2e+140) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 5d+40) then
        tmp = x * x
    else if ((z * z) <= 2d+61) then
        tmp = (-4.0d0) * ((z * z) * y)
    else if ((z * z) <= 2d+140) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+40) {
		tmp = x * x;
	} else if ((z * z) <= 2e+61) {
		tmp = -4.0 * ((z * z) * y);
	} else if ((z * z) <= 2e+140) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+40:
		tmp = x * x
	elif (z * z) <= 2e+61:
		tmp = -4.0 * ((z * z) * y)
	elif (z * z) <= 2e+140:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+40)
		tmp = Float64(x * x);
	elseif (Float64(z * z) <= 2e+61)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	elseif (Float64(z * z) <= 2e+140)
		tmp = Float64(x * x);
	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) <= 5e+40)
		tmp = x * x;
	elseif ((z * z) <= 2e+61)
		tmp = -4.0 * ((z * z) * y);
	elseif ((z * z) <= 2e+140)
		tmp = x * x;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+40], N[(x * x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+61], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+140], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000003e40 or 1.9999999999999999e61 < (*.f64 z z) < 2.00000000000000012e140

   1. Initial program 95.1%

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

      1. associate-*l* 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in x around inf 58.8%

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

      1. unpow2 58.8%

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

   6. Simplified 58.8%

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

   if 5.00000000000000003e40 < (*.f64 z z) < 1.9999999999999999e61

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l* 100.0%

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

      7. add-cube-cbrt 99.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \sqrt[3]{y \cdot -4}\right)} \cdot \left(z \cdot z - t\right) + x \cdot x \]

      8. associate-*l* 99.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \left(\sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right)\right)} + x \cdot x \]

      9. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)} \]

      10. pow2 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{-4 \cdot \left({1}^{0.3333333333333333} \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if 2.00000000000000012e140 < (*.f64 z z)

   1. Initial program 83.8%

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

      1. associate-*l* 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in z around inf 75.8%

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

      1. *-commutative 75.8%

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

      2. unpow2 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-*r* 75.8%

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

      5. associate-*l* 81.0%

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

   6. Simplified 81.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+61}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+140}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 6: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+37}:\\ \;\;\;\;x \cdot x - \left(y \cdot -4\right) \cdot t\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+269}:\\ \;\;\;\;x \cdot x - y \cdot \left(\left(z \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+37)
   (- (* x x) (* (* y -4.0) t))
   (if (<= (* z z) 5e+269)
     (- (* x x) (* y (* (* z z) 4.0)))
     (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+37) {
		tmp = (x * x) - ((y * -4.0) * t);
	} else if ((z * z) <= 5e+269) {
		tmp = (x * x) - (y * ((z * z) * 4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+37) then
        tmp = (x * x) - ((y * (-4.0d0)) * t)
    else if ((z * z) <= 5d+269) then
        tmp = (x * x) - (y * ((z * z) * 4.0d0))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+37) {
		tmp = (x * x) - ((y * -4.0) * t);
	} else if ((z * z) <= 5e+269) {
		tmp = (x * x) - (y * ((z * z) * 4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+37:
		tmp = (x * x) - ((y * -4.0) * t)
	elif (z * z) <= 5e+269:
		tmp = (x * x) - (y * ((z * z) * 4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+37)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * -4.0) * t));
	elseif (Float64(z * z) <= 5e+269)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(Float64(z * z) * 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 * z) <= 1e+37)
		tmp = (x * x) - ((y * -4.0) * t);
	elseif ((z * z) <= 5e+269)
		tmp = (x * x) - (y * ((z * z) * 4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+37], N[(N[(x * x), $MachinePrecision] - N[(N[(y * -4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+269], N[(N[(x * x), $MachinePrecision] - N[(y * N[(N[(z * z), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 9.99999999999999954e36

   1. Initial program 95.4%

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

      1. associate-*l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 92.5%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

   6. Simplified 91.8%

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

   if 9.99999999999999954e36 < (*.f64 z z) < 5.0000000000000002e269

   1. Initial program 95.7%

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

      1. associate-*l* 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in z around inf 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

   if 5.0000000000000002e269 < (*.f64 z z)

   1. Initial program 79.1%

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

      1. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in z around inf 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*r* 82.9%

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

      5. associate-*l* 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+37}:\\ \;\;\;\;x \cdot x - \left(y \cdot -4\right) \cdot t\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+269}:\\ \;\;\;\;x \cdot x - y \cdot \left(\left(z \cdot z\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 7: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+269}:\\ \;\;\;\;x \cdot x + y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+269)
   (+ (* x x) (* 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) <= 5e+269) {
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 5d+269) then
        tmp = (x * x) + (y * (4.0d0 * (t - (z * z))))
    else
        tmp = (x * x) - (z * (z * (y * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+269) {
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+269:
		tmp = (x * x) + (y * (4.0 * (t - (z * z))))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+269)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(4.0 * Float64(t - Float64(z * z)))));
	else
		tmp = Float64(Float64(x * x) - 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) <= 5e+269)
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+269], N[(N[(x * x), $MachinePrecision] + N[(y * N[(4.0 * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000002e269

   1. Initial program 95.5%

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

      1. associate-*l* 96.0%

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

   3. Simplified 96.0%

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

   if 5.0000000000000002e269 < (*.f64 z z)

   1. Initial program 79.1%

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

      1. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in z around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-*r* 79.1%

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

      5. associate-*l* 93.5%

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

      6. *-commutative 93.5%

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

   6. Simplified 93.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+269}:\\ \;\;\;\;x \cdot x + y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 8: 89.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+37)
   (- (* x x) (* (* y -4.0) t))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+37) {
		tmp = (x * x) - ((y * -4.0) * t);
	} else {
		tmp = (x * x) - (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+37) then
        tmp = (x * x) - ((y * (-4.0d0)) * t)
    else
        tmp = (x * x) - (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+37) {
		tmp = (x * x) - ((y * -4.0) * t);
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+37:
		tmp = (x * x) - ((y * -4.0) * t)
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+37)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * -4.0) * t));
	else
		tmp = Float64(Float64(x * x) - 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+37)
		tmp = (x * x) - ((y * -4.0) * t);
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+37], N[(N[(x * x), $MachinePrecision] - N[(N[(y * -4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999954e36

   1. Initial program 95.4%

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

      1. associate-*l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 92.5%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

   6. Simplified 91.8%

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

   if 9.99999999999999954e36 < (*.f64 z z)

   1. Initial program 85.3%

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

      1. associate-*l* 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in z around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-*r* 80.2%

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

      5. associate-*l* 89.2%

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

      6. *-commutative 89.2%

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

   6. Simplified 89.2%

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

4. Final simplification 90.5%

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

Alternative 9: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{+126}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.25e+126) (* (* y -4.0) (- (* z z) t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.25e+126) {
		tmp = (y * -4.0) * ((z * z) - 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) <= 1.25d+126) then
        tmp = (y * (-4.0d0)) * ((z * z) - 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) <= 1.25e+126) {
		tmp = (y * -4.0) * ((z * z) - t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.25e+126:
		tmp = (y * -4.0) * ((z * z) - t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.25e+126)
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - 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) <= 1.25e+126)
		tmp = (y * -4.0) * ((z * z) - t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.25e+126], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.24999999999999994e126

   1. Initial program 95.5%

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

      1. associate-*l* 95.5%

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

   3. Simplified 95.5%

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

      1. sub-neg 95.5%

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

      2. +-commutative 95.5%

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

      3. distribute-rgt-neg-in 95.5%

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

      4. distribute-lft-neg-in 95.5%

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

      5. metadata-eval 95.5%

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

      6. associate-*l* 95.5%

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

      7. add-cube-cbrt 94.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \sqrt[3]{y \cdot -4}\right)} \cdot \left(z \cdot z - t\right) + x \cdot x \]

      8. associate-*l* 94.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}\right) \cdot \left(\sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right)\right)} + x \cdot x \]

      9. fma-def 94.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot -4} \cdot \sqrt[3]{y \cdot -4}, \sqrt[3]{y \cdot -4} \cdot \left(z \cdot z - t\right), x \cdot x\right)} \]

      10. pow2 94.7%

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

   5. Applied egg-rr 94.7%

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

   6. Taylor expanded in x around 0 83.8%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\left({z}^{2} - t\right) \cdot y\right) \cdot {1}^{0.3333333333333333}\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 83.8%

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

      2. *-rgt-identity 83.8%

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

      3. *-commutative 83.8%

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

      4. unpow2 83.8%

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

      5. distribute-rgt-out-- 81.1%

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

      6. distribute-lft-out-- 80.4%

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

      7. *-commutative 80.4%

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

      8. associate-*r* 80.4%

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

      9. *-commutative 80.4%

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

      10. *-commutative 80.4%

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

      11. associate-*r* 80.4%

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

      12. *-commutative 80.4%

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

      13. distribute-lft-out-- 83.8%

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

   8. Simplified 83.8%

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

   if 1.24999999999999994e126 < (*.f64 x x)

   1. Initial program 83.8%

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

      1. associate-*l* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in x around inf 79.5%

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

      1. unpow2 79.5%

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

   6. Simplified 79.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{+126}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10: 45.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-28}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 8.5e-28) (* t (* y 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 8.5e-28) {
		tmp = t * (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 <= 8.5d-28) then
        tmp = t * (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 <= 8.5e-28) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 8.5e-28:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 8.5e-28)
		tmp = Float64(t * 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 <= 8.5e-28)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 8.5e-28], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999925e-28

   1. Initial program 91.4%

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

      1. associate-*l* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in t around inf 33.4%

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

      1. associate-*r* 33.4%

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

   6. Simplified 33.4%

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

   if 8.49999999999999925e-28 < x

   1. Initial program 88.0%

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

      1. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around inf 63.9%

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

      1. unpow2 63.9%

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

   6. Simplified 63.9%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-28}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 11: 40.9% 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. Step-by-step derivation

   1. associate-*l* 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in x around inf 42.3%

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

   1. unpow2 42.3%

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

6. Simplified 42.3%

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

7. Final simplification 42.3%

   \[\leadsto x \cdot x \]

Developer target: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023223 
(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))))