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

Percentage Accurate: 90.4% → 92.7%

Time: 12.2s

Alternatives: 7

Speedup: 0.4×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% 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: 92.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z\_m \cdot z\_m, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2e+143)
   (fma (* y 4.0) (- t (* z_m z_m)) (* x x))
   (* (* z_m z_m) (* y -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+143) {
		tmp = fma((y * 4.0), (t - (z_m * z_m)), (x * x));
	} else {
		tmp = (z_m * z_m) * (y * -4.0);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2e+143)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z_m * z_m)), Float64(x * x));
	else
		tmp = Float64(Float64(z_m * z_m) * Float64(y * -4.0));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2e+143], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2e143

   1. Initial program 93.6%

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

      1. cancel-sign-sub-inv 93.6%

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

      2. distribute-lft-neg-out 93.6%

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

      3. +-commutative 93.6%

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

      4. associate-*l* 93.6%

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

      5. distribute-lft-neg-in 93.6%

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

      6. associate-*l* 93.6%

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

      7. distribute-rgt-neg-in 93.6%

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

      8. fma-define 96.4%

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

      9. sub-neg 96.4%

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

      10. +-commutative 96.4%

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

      11. distribute-neg-in 96.4%

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

      12. remove-double-neg 96.4%

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

      13. sub-neg 96.4%

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

   3. Simplified 96.4%

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

   if 2e143 < z

   1. Initial program 73.7%

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

      1. fma-neg 81.2%

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

      2. distribute-lft-neg-in 81.2%

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

      3. *-commutative 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

      5. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in z around inf 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

   7. Simplified 81.2%

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

      1. unpow2 81.2%

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

   9. Applied egg-rr 81.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.7% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (fma x x (* (* y -4.0) (- (* z_m z_m) t))))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $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 93.2%

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

   2. distribute-lft-neg-in 93.2%

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

   3. *-commutative 93.2%

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

   4. distribute-rgt-neg-in 93.2%

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

   5. metadata-eval 93.2%

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

3. Simplified 93.2%

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

5. Final simplification 93.2%

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

Alternative 3: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z\_m \cdot z\_m\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z_m z_m))))))
   (if (<= t_1 INFINITY) t_1 (* y (* t -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z_m * z_m)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y * (t * -4.0)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z_m * z_m))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(t * -4.0));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y * (t * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.7%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 21.4%

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

      4. swap-sqr 21.4%

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

      5. metadata-eval 21.4%

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

      6. metadata-eval 21.4%

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

      7. swap-sqr 21.4%

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

      8. sqrt-unprod 21.4%

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

      9. add-sqr-sqrt 50.0%

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

      10. add-cube-cbrt 50.0%

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

      11. pow3 50.0%

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

   4. Applied egg-rr 0.0%

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

      1. rem-cbrt-cube 0.0%

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

      2. *-un-lft-identity 0.0%

         \[\leadsto x \cdot x - {\color{blue}{\left(1 \cdot \sqrt[3]{{\left(\sqrt[3]{y \cdot \left(4 \cdot \left({z}^{2} - t\right)\right)}\right)}^{3}}\right)}}^{3} \]

      3. unpow3 0.0%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{y \cdot \left(4 \cdot \left({z}^{2} - t\right)\right)} \cdot \sqrt[3]{y \cdot \left(4 \cdot \left({z}^{2} - t\right)\right)}\right) \cdot \sqrt[3]{y \cdot \left(4 \cdot \left({z}^{2} - t\right)\right)}}}\right)}^{3} \]

      4. add-cube-cbrt 0.0%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{\color{blue}{y \cdot \left(4 \cdot \left({z}^{2} - t\right)\right)}}\right)}^{3} \]

      5. unpow2 0.0%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \left(\color{blue}{z \cdot z} - t\right)\right)}\right)}^{3} \]

      6. fma-neg 0.0%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \color{blue}{\mathsf{fma}\left(z, z, -t\right)}\right)}\right)}^{3} \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right)\right)}\right)}^{3} \]

      8. sqrt-unprod 21.4%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}\right)\right)}\right)}^{3} \]

      9. sqr-neg 21.4%

         \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, \sqrt{\color{blue}{t \cdot t}}\right)\right)}\right)}^{3} \]

      10. sqrt-unprod 21.4%

          \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\right)}\right)}^{3} \]

      11. add-sqr-sqrt 35.7%

          \[\leadsto x \cdot x - {\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, \color{blue}{t}\right)\right)}\right)}^{3} \]

   6. Applied egg-rr 35.7%

      \[\leadsto x \cdot x - {\color{blue}{\left(1 \cdot \sqrt[3]{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, t\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. *-lft-identity 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in t around inf 51.7%

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(y \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 51.7%

          \[\leadsto \color{blue}{\left(-1 \cdot t\right) \cdot \left(y \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)} \]

       2. rem-cube-cbrt 51.7%

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

       3. neg-mul-1 51.7%

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

       4. distribute-lft-neg-in 51.7%

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

       5. associate-*r* 51.7%

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

       6. *-commutative 51.7%

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

       7. distribute-rgt-neg-in 51.7%

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

       8. metadata-eval 51.7%

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

       9. associate-*r* 51.7%

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

   11. Simplified 51.7%

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

4. Final simplification 93.3%

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

Alternative 4: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z\_m \leq 3.1 \cdot 10^{+42}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2.3e-95)
   (* 4.0 (* y t))
   (if (<= z_m 3.1e+42) (* x x) (* (* z_m z_m) (* y -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-95) {
		tmp = 4.0 * (y * t);
	} else if (z_m <= 3.1e+42) {
		tmp = x * x;
	} else {
		tmp = (z_m * z_m) * (y * -4.0);
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.3d-95) then
        tmp = 4.0d0 * (y * t)
    else if (z_m <= 3.1d+42) then
        tmp = x * x
    else
        tmp = (z_m * z_m) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-95) {
		tmp = 4.0 * (y * t);
	} else if (z_m <= 3.1e+42) {
		tmp = x * x;
	} else {
		tmp = (z_m * z_m) * (y * -4.0);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 2.3e-95:
		tmp = 4.0 * (y * t)
	elif z_m <= 3.1e+42:
		tmp = x * x
	else:
		tmp = (z_m * z_m) * (y * -4.0)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.3e-95)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (z_m <= 3.1e+42)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(z_m * z_m) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.3e-95)
		tmp = 4.0 * (y * t);
	elseif (z_m <= 3.1e+42)
		tmp = x * x;
	else
		tmp = (z_m * z_m) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 2.3e-95], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.1e+42], N[(x * x), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.29999999999999999e-95

   1. Initial program 93.8%

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

      1. fma-neg 96.3%

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

      2. distribute-lft-neg-in 96.3%

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

      3. *-commutative 96.3%

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

      4. distribute-rgt-neg-in 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t around inf 39.6%

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

      1. *-commutative 39.6%

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

   7. Simplified 39.6%

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

   if 2.29999999999999999e-95 < z < 3.1000000000000002e42

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 93.9%

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

   5. Simplified 46.7%

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

      1. --rgt-identity 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 3.1000000000000002e42 < z

   1. Initial program 80.6%

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

      1. fma-neg 85.3%

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

      2. distribute-lft-neg-in 85.3%

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

      3. *-commutative 85.3%

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

      4. distribute-rgt-neg-in 85.3%

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

      5. metadata-eval 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in z around inf 73.6%

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

      1. associate-*r* 73.6%

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

      2. *-commutative 73.6%

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

   7. Simplified 73.6%

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

      1. unpow2 73.6%

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

   9. Applied egg-rr 73.6%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+42}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 7.2e+54) (- (* x x) (* y (* t -4.0))) (* (* z_m z_m) (* y -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 7.2e+54) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z_m * z_m) * (y * -4.0);
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 7.2d+54) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (z_m * z_m) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 7.2e+54) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z_m * z_m) * (y * -4.0);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 7.2e+54:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (z_m * z_m) * (y * -4.0)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 7.2e+54)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(z_m * z_m) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 7.2e+54)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (z_m * z_m) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 7.2e+54], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.2000000000000003e54

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0 74.7%

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

      1. *-commutative 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-*l* 74.7%

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

   5. Simplified 74.7%

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

   if 7.2000000000000003e54 < z

   1. Initial program 80.0%

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

      1. fma-neg 84.8%

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

      2. distribute-lft-neg-in 84.8%

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

      3. *-commutative 84.8%

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

      4. distribute-rgt-neg-in 84.8%

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

      5. metadata-eval 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in z around inf 74.3%

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

      1. associate-*r* 74.3%

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

      2. *-commutative 74.3%

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

   7. Simplified 74.3%

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

      1. unpow2 74.3%

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

   9. Applied egg-rr 74.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.9 \cdot 10^{+51}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* x x) 1.9e+51) (* 4.0 (* y t)) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 1.9e+51) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1.9d+51) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 1.9e+51) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (x * x) <= 1.9e+51:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.9e+51)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((x * x) <= 1.9e+51)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.9e+51], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.8999999999999999e51

   1. Initial program 93.2%

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

      1. fma-neg 93.2%

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

      2. distribute-lft-neg-in 93.2%

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

      3. *-commutative 93.2%

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

      4. distribute-rgt-neg-in 93.2%

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

      5. metadata-eval 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in t around inf 49.1%

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

      1. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   if 1.8999999999999999e51 < (*.f64 x x)

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0 86.4%

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

   5. Simplified 69.7%

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

      1. --rgt-identity 69.7%

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

   7. Applied egg-rr 69.7%

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

Alternative 7: 41.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t) :precision binary64 (* x x))

C

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

Fortran

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

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return x * x;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return x * x

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, 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. Taylor expanded in y around 0 90.5%

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

5. Simplified 35.3%

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

   1. --rgt-identity 35.3%

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

7. Applied egg-rr 35.3%

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

Developer Target 1: 90.4% 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 2024135 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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