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

Percentage Accurate: 91.4% → 94.1%

Time: 13.0s

Alternatives: 8

Speedup: 0.5×

• 49.3% 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: 91.4% 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: 94.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* (- (* z z) t) (* y 4.0)) 2e+305)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* t (* y (+ 4.0 (* -4.0 (* z (/ z t))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z * (z / t)))));
	}
	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) - t) * (y * 4.0d0)) <= 2d+305) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = t * (y * (4.0d0 + ((-4.0d0) * (z * (z / t)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z * (z / t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((z * z) - t) * (y * 4.0)) <= 2e+305:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = t * (y * (4.0 + (-4.0 * (z * (z / t)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(z * z) - t) * Float64(y * 4.0)) <= 2e+305)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(-4.0 * Float64(z * Float64(z / t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = t * (y * (4.0 + (-4.0 * (z * (z / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 2e+305], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(-4.0 * N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 1.9999999999999999e305

   1. Initial program 98.2%

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

   if 1.9999999999999999e305 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 59.8%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. *-commutative 59.8%

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

      2. fma-define 59.8%

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

      3. associate-*r/ 59.8%

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

      4. *-commutative 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-*r* 59.8%

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

      7. associate-/l* 59.6%

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

      8. *-commutative 59.6%

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

   5. Simplified 59.6%

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

   6. Taylor expanded in y around -inf 69.6%

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

      1. unpow2 69.6%

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

      2. *-un-lft-identity 69.6%

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

      3. times-frac 73.3%

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

   8. Applied egg-rr 73.3%

      \[\leadsto t \cdot \left(y \cdot \left(4 + -4 \cdot \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{t}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

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

Alternative 2: 93.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $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. fma-neg 92.5%

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

   2. distribute-lft-neg-in 92.5%

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

   3. *-commutative 92.5%

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

   4. distribute-rgt-neg-in 92.5%

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

   5. metadata-eval 92.5%

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

3. Simplified 92.5%

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

Alternative 3: 94.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* (- (* z z) t) (* y 4.0)) 2e+305)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* t (* y (+ 4.0 (* -4.0 (/ z (/ t z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	}
	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) - t) * (y * 4.0d0)) <= 2d+305) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = t * (y * (4.0d0 + ((-4.0d0) * (z / (t / z)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((z * z) - t) * (y * 4.0)) <= 2e+305:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(z * z) - t) * Float64(y * 4.0)) <= 2e+305)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(-4.0 * Float64(z / Float64(t / z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((z * z) - t) * (y * 4.0)) <= 2e+305)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 2e+305], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(-4.0 * N[(z / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 1.9999999999999999e305

   1. Initial program 98.2%

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

   if 1.9999999999999999e305 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 59.8%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. *-commutative 59.8%

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

      2. fma-define 59.8%

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

      3. associate-*r/ 59.8%

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

      4. *-commutative 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-*r* 59.8%

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

      7. associate-/l* 59.6%

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

      8. *-commutative 59.6%

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

   5. Simplified 59.6%

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

   6. Taylor expanded in y around -inf 69.6%

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

      1. unpow2 69.6%

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

      2. *-un-lft-identity 69.6%

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

      3. times-frac 73.3%

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

   8. Applied egg-rr 73.3%

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

      1. /-rgt-identity 73.3%

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

      2. clear-num 73.3%

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

      3. un-div-inv 73.3%

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

   10. Applied egg-rr 73.3%

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

4. Final simplification 93.2%

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

Alternative 4: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+66}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + -4 \cdot \frac{z}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 8.2e+66)
   (- (* x x) (* -4.0 (* t y)))
   (* t (* y (+ 4.0 (* -4.0 (/ z (/ t z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.2e+66) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	}
	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 <= 8.2d+66) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = t * (y * (4.0d0 + ((-4.0d0) * (z / (t / z)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8.2e+66) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 8.2e+66:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8.2e+66)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(-4.0 * Float64(z / Float64(t / z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8.2e+66)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = t * (y * (4.0 + (-4.0 * (z / (t / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 8.2e+66], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(4.0 + N[(-4.0 * N[(z / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.19999999999999989e66

   1. Initial program 93.4%

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

   3. Taylor expanded in z around 0 79.1%

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

      1. *-commutative 79.1%

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

   5. Simplified 79.1%

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

   if 8.19999999999999989e66 < z

   1. Initial program 77.5%

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

   3. Taylor expanded in t around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. fma-define 73.4%

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

      3. associate-*r/ 73.4%

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

      4. *-commutative 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*r* 73.4%

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

      7. associate-/l* 69.2%

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

      8. *-commutative 69.2%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in y around -inf 77.6%

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

      1. unpow2 77.6%

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

      2. *-un-lft-identity 77.6%

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

      3. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. /-rgt-identity 79.5%

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

      2. clear-num 79.5%

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

      3. un-div-inv 79.5%

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

   10. Applied egg-rr 79.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+66}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + -4 \cdot \frac{z}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.5e+31) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.5e+31) {
		tmp = y * (t * 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 <= 2.5d+31) then
        tmp = y * (t * 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 <= 2.5e+31) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.5e+31:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.5e+31)
		tmp = Float64(y * Float64(t * 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 <= 2.5e+31)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.5e+31], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000013e31

   1. Initial program 92.6%

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

   3. Taylor expanded in t around inf 40.6%

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

      1. associate-*r* 40.6%

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

   5. Simplified 40.6%

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

   if 2.50000000000000013e31 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in y around 0 83.8%

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

   5. Simplified 72.6%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.35e+30) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.35e+30) {
		tmp = y * (t * 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 <= 2.35d+30) then
        tmp = y * (t * 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 <= 2.35e+30) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.35e+30:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.35e+30)
		tmp = Float64(y * Float64(t * 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 <= 2.35e+30)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.35e+30], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.34999999999999995e30

   1. Initial program 92.6%

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

   3. Taylor expanded in t around inf 40.6%

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

      1. associate-*r* 40.6%

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

   5. Simplified 40.6%

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

   if 2.34999999999999995e30 < x

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. distribute-lft-neg-in 83.8%

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

      3. *-commutative 83.8%

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

      4. distribute-lft-neg-in 83.8%

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

      5. metadata-eval 83.8%

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

      6. *-commutative 83.8%

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

      7. add-sqr-sqrt 44.3%

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

      8. sqrt-unprod 67.6%

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

      9. swap-sqr 67.6%

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

      10. metadata-eval 67.6%

          \[\leadsto x \cdot x + \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left(z \cdot z - t\right) \]

      11. metadata-eval 67.6%

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

      12. swap-sqr 67.6%

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

      13. sqrt-unprod 34.4%

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

      14. add-sqr-sqrt 65.6%

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

      15. expm1-log1p-u 62.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      16. add-sqr-sqrt 32.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

   4. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

   6. Simplified 69.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 72.6%

         \[\leadsto \color{blue}{{x}^{2}} \]

      2. unpow2 72.6%

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

   8. Applied egg-rr 72.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) * (t * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in z around 0 69.1%

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

   1. *-commutative 69.1%

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

5. Simplified 69.1%

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

6. Final simplification 69.1%

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

Alternative 8: 41.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. sub-neg 90.5%

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

   2. distribute-lft-neg-in 90.5%

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

   3. *-commutative 90.5%

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

   4. distribute-lft-neg-in 90.5%

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

   5. metadata-eval 90.5%

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

   6. *-commutative 90.5%

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

   7. add-sqr-sqrt 47.7%

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

   8. sqrt-unprod 57.3%

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

   9. swap-sqr 57.3%

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

   10. metadata-eval 57.3%

       \[\leadsto x \cdot x + \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left(z \cdot z - t\right) \]

   11. metadata-eval 57.3%

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

   12. swap-sqr 57.3%

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

   13. sqrt-unprod 18.4%

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

   14. add-sqr-sqrt 37.4%

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

   15. expm1-log1p-u 35.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

   16. add-sqr-sqrt 17.6%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

4. Applied egg-rr 66.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

6. Simplified 40.5%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 42.0%

      \[\leadsto \color{blue}{{x}^{2}} \]

   2. unpow2 42.0%

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

8. Applied egg-rr 42.0%

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

Developer target: 91.4% 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 2024103 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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