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

Percentage Accurate: 90.7% → 94.1%

Time: 11.6s

Alternatives: 7

Speedup: 0.7×

• 49.2% 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: 94.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + -4 \cdot \left(z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+291)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* 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) <= 2e+291) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} 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(z * z) <= 2e+291)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(-4.0 * Float64(z * Float64(z / t))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+291], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $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 z z) < 1.9999999999999999e291

   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. cancel-sign-sub-inv 96.2%

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

      2. distribute-lft-neg-out 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-*l* 96.2%

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

      5. distribute-lft-neg-in 96.2%

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

      6. associate-*l* 96.2%

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

      7. distribute-rgt-neg-in 96.2%

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

      8. fma-define 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. remove-double-neg 99.4%

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

      13. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   if 1.9999999999999999e291 < (*.f64 z z)

   1. Initial program 70.7%

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

   3. Taylor expanded in t around inf 70.7%

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

      1. +-commutative 70.7%

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

      2. fma-define 70.7%

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

      3. associate-/l* 70.7%

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

      4. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in y around -inf 76.5%

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

      1. unpow2 76.5%

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

      2. *-un-lft-identity 76.5%

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

      3. times-frac 83.3%

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

   8. Applied egg-rr 83.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + -4 \cdot \left(z \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.6e+174) (fma x x (* (- (* z z) t) (* y -4.0))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.6e+174) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.6e+174)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 3.6e+174], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000002e174

   1. Initial program 90.4%

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

      1. fma-neg 92.6%

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

      2. distribute-lft-neg-in 92.6%

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

      3. *-commutative 92.6%

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

      4. distribute-rgt-neg-in 92.6%

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

      5. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

   if 3.6000000000000002e174 < x

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0 77.3%

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

   5. Simplified 100.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ t_2 := y \cdot \left(4 \cdot t\right)\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-256}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1020000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* (* z z) -4.0))) (t_2 (* y (* 4.0 t))))
   (if (<= x 1.25e-256)
     t_2
     (if (<= x 2.25e-29)
       t_1
       (if (<= x 1020000.0) t_2 (if (<= x 6.5e+16) t_1 (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z * z) * -4.0);
	double t_2 = y * (4.0 * t);
	double tmp;
	if (x <= 1.25e-256) {
		tmp = t_2;
	} else if (x <= 2.25e-29) {
		tmp = t_1;
	} else if (x <= 1020000.0) {
		tmp = t_2;
	} else if (x <= 6.5e+16) {
		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) :: t_2
    real(8) :: tmp
    t_1 = y * ((z * z) * (-4.0d0))
    t_2 = y * (4.0d0 * t)
    if (x <= 1.25d-256) then
        tmp = t_2
    else if (x <= 2.25d-29) then
        tmp = t_1
    else if (x <= 1020000.0d0) then
        tmp = t_2
    else if (x <= 6.5d+16) 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 = y * ((z * z) * -4.0);
	double t_2 = y * (4.0 * t);
	double tmp;
	if (x <= 1.25e-256) {
		tmp = t_2;
	} else if (x <= 2.25e-29) {
		tmp = t_1;
	} else if (x <= 1020000.0) {
		tmp = t_2;
	} else if (x <= 6.5e+16) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * ((z * z) * -4.0)
	t_2 = y * (4.0 * t)
	tmp = 0
	if x <= 1.25e-256:
		tmp = t_2
	elif x <= 2.25e-29:
		tmp = t_1
	elif x <= 1020000.0:
		tmp = t_2
	elif x <= 6.5e+16:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z * z) * -4.0))
	t_2 = Float64(y * Float64(4.0 * t))
	tmp = 0.0
	if (x <= 1.25e-256)
		tmp = t_2;
	elseif (x <= 2.25e-29)
		tmp = t_1;
	elseif (x <= 1020000.0)
		tmp = t_2;
	elseif (x <= 6.5e+16)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * ((z * z) * -4.0);
	t_2 = y * (4.0 * t);
	tmp = 0.0;
	if (x <= 1.25e-256)
		tmp = t_2;
	elseif (x <= 2.25e-29)
		tmp = t_1;
	elseif (x <= 1020000.0)
		tmp = t_2;
	elseif (x <= 6.5e+16)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25e-256], t$95$2, If[LessEqual[x, 2.25e-29], t$95$1, If[LessEqual[x, 1020000.0], t$95$2, If[LessEqual[x, 6.5e+16], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25e-256 or 2.2499999999999999e-29 < x < 1.02e6

   1. Initial program 92.9%

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

   3. Taylor expanded in t around inf 32.3%

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

      1. associate-*r* 32.3%

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

   5. Simplified 32.3%

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

   if 1.25e-256 < x < 2.2499999999999999e-29 or 1.02e6 < x < 6.5e16

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf 55.7%

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

      1. associate-*r* 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*l* 55.7%

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

   5. Simplified 55.7%

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

      1. unpow2 55.7%

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

   7. Applied egg-rr 55.7%

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

   if 6.5e16 < x

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0 77.2%

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

   5. Simplified 77.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-256}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{elif}\;x \leq 1020000:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7 \cdot 10^{+189}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 7e+189) (* t (* y (+ 4.0 (/ (* z -4.0) (/ t z))))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7e+189) {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	} 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) <= 7d+189) then
        tmp = t * (y * (4.0d0 + ((z * (-4.0d0)) / (t / z))))
    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) <= 7e+189) {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 7e+189:
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 7e+189)
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(Float64(z * -4.0) / Float64(t / z)))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 7e+189)
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 7e+189], N[(t * N[(y * N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.99999999999999991e189

   1. Initial program 95.3%

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

   3. Taylor expanded in t around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-define 90.0%

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

      3. associate-/l* 88.8%

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

      4. *-commutative 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in y around -inf 77.5%

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

      1. unpow2 77.5%

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

      2. *-un-lft-identity 77.5%

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

      3. times-frac 80.4%

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

   8. Applied egg-rr 80.4%

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

      1. /-rgt-identity 80.4%

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

      2. associate-*r* 80.4%

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

      3. clear-num 80.4%

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

      4. un-div-inv 80.4%

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

   10. Applied egg-rr 80.4%

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

   if 6.99999999999999991e189 < (*.f64 x x)

   1. Initial program 79.0%

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

   3. Taylor expanded in y around 0 79.0%

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

   5. Simplified 82.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7 \cdot 10^{+189}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+142}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.5e+142) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.5e+142) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 7.5d+142) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.5e+142) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 7.5e+142:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.5e+142)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 7.5e+142)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.5e+142], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5000000000000002e142

   1. Initial program 92.3%

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

   if 7.5000000000000002e142 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in y around 0 67.7%

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

   5. Simplified 90.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+142}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.1e+34) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.1e+34) {
		tmp = y * (4.0 * 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 <= 2.1d+34) then
        tmp = y * (4.0d0 * 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 <= 2.1e+34) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.1e+34:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.1e+34)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.1e+34)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.1e+34], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000017e34

   1. Initial program 92.8%

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

   3. Taylor expanded in t around inf 33.7%

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

      1. associate-*r* 33.7%

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

   5. Simplified 33.7%

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

   if 2.10000000000000017e34 < x

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0 75.9%

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

   5. Simplified 79.7%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return y * (4.0 * 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 = y * (4.0d0 * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in t around inf 29.1%

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

   1. associate-*r* 29.1%

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

5. Simplified 29.1%

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

6. Final simplification 29.1%

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

Developer target: 90.6% 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 2024053 
(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))))