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

Percentage Accurate: 90.3% → 97.7%

Time: 4.9s

Alternatives: 5

Speedup: 1.0×

• 48.6% 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: 90.3% 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: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(z\_m \cdot z\_m - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot z\_m, y \cdot z\_m, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 5e+153)
   (fma (- (* z_m z_m) t) (* -4.0 y) (* x x))
   (fma (* -4.0 z_m) (* y z_m) (* x x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5e+153) {
		tmp = fma(((z_m * z_m) - t), (-4.0 * y), (x * x));
	} else {
		tmp = fma((-4.0 * z_m), (y * z_m), (x * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 5e+153)
		tmp = fma(Float64(Float64(z_m * z_m) - t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = fma(Float64(-4.0 * z_m), Float64(y * z_m), Float64(x * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 5e+153], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z$95$m), $MachinePrecision] * N[(y * z$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000018e153

   1. Initial program 93.8%

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

   4. Applied rewrites 94.6%

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

   if 5.00000000000000018e153 < z

   1. Initial program 60.8%

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

   3. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   7. Applied rewrites 90.8%

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

Alternative 2: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot z\_m, y \cdot z\_m, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 0.0055)
   (fma (* t 4.0) y (* x x))
   (fma (* -4.0 z_m) (* y z_m) (* x x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 0.0055) {
		tmp = fma((t * 4.0), y, (x * x));
	} else {
		tmp = fma((-4.0 * z_m), (y * z_m), (x * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 0.0055)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	else
		tmp = fma(Float64(-4.0 * z_m), Float64(y * z_m), Float64(x * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 0.0055], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z$95$m), $MachinePrecision] * N[(y * z$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0054999999999999997

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

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

      16. fp-cancel-sign-sub N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. distribute-lft-neg-in N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. lower-fma.f64 N/A

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

      24. lower-*.f64 N/A

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

      25. unpow2 N/A

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 76.9%

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

   if 0.0054999999999999997 < z

   1. Initial program 81.4%

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

   3. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 88.4%

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

Alternative 3: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.4e+20) (fma (* t 4.0) y (* x x)) (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.4e+20) {
		tmp = fma((t * 4.0), y, (x * x));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.4e+20)
		tmp = fma(Float64(t * 4.0), y, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -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, 1.4e+20], N[(N[(t * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e20

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

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

      16. fp-cancel-sign-sub N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. distribute-lft-neg-in N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

      23. lower-fma.f64 N/A

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

      24. lower-*.f64 N/A

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

      25. unpow2 N/A

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 76.5%

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

   if 1.4e20 < z

   1. Initial program 79.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 70.0%

      \[\leadsto \left(\left(y \cdot z\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 60.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0055:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 0.0055) (* (* t 4.0) y) (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 0.0055) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = ((y * z_m) * z_m) * -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 <= 0.0055d0) then
        tmp = (t * 4.0d0) * y
    else
        tmp = ((y * z_m) * z_m) * (-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 <= 0.0055) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 0.0055:
		tmp = (t * 4.0) * y
	else:
		tmp = ((y * z_m) * z_m) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 0.0055)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -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 <= 0.0055)
		tmp = (t * 4.0) * y;
	else
		tmp = ((y * z_m) * z_m) * -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, 0.0055], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0054999999999999997

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   if 0.0054999999999999997 < z

   1. Initial program 81.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   7. Applied rewrites 67.2%

      \[\leadsto \left(\left(y \cdot z\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 32.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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 = (t * 4.0d0) * y
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return (t * 4.0) * y;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return (t * 4.0) * y

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 34.2

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

5. Applied rewrites 34.2%

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

7. Applied rewrites 34.2%

   \[\leadsto \left(t \cdot 4\right) \cdot \color{blue}{y} \]
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 2024342 
(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))))