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

Percentage Accurate: 91.5% → 96.5%

Time: 9.8s

Alternatives: 8

Speedup: 0.6×

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% 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: 96.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z\_m \cdot z\_m, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 2.8e+149)
   (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 <= 2.8e+149) {
		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 <= 2.8e+149)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z_m * z_m)), Float64(x * x));
	else
		tmp = Float64(z_m * Float64(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, 2.8e+149], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.7999999999999999e149

   1. Initial program 94.2%

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

      1. cancel-sign-sub-inv 94.2%

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

      2. distribute-lft-neg-out 94.2%

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

      3. +-commutative 94.2%

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

      4. distribute-lft-neg-out 94.2%

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

      5. distribute-lft-neg-in 94.2%

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

      6. distribute-rgt-neg-in 94.2%

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

      7. fma-define 96.9%

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

      8. sub-neg 96.9%

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

      9. +-commutative 96.9%

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

      10. distribute-neg-in 96.9%

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

      11. remove-double-neg 96.9%

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

      12. sub-neg 96.9%

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

   3. Simplified 96.9%

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

   if 2.7999999999999999e149 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in z around inf 86.7%

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

      1. associate-*r* 86.7%

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

      2. *-commutative 86.7%

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

   5. Simplified 86.7%

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

      1. unpow2 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. add-sqr-sqrt 29.2%

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

      2. pow2 29.2%

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

      3. sqrt-prod 29.2%

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

      4. sqrt-prod 34.5%

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

      5. add-sqr-sqrt 34.5%

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

      6. *-commutative 34.5%

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

   9. Applied egg-rr 34.5%

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

       1. *-commutative 34.5%

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

       2. unpow-prod-down 29.2%

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

       3. pow2 29.2%

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

       4. add-sqr-sqrt 86.7%

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

       5. *-commutative 86.7%

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

       6. pow2 86.7%

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

       7. associate-*r* 94.5%

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

   11. Applied egg-rr 94.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z\_m \cdot z\_m - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= x 1.05e+170) (fma x x (* (- (* z_m z_m) t) (* y -4.0))) (pow x 2.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (x <= 1.05e+170) {
		tmp = fma(x, x, (((z_m * z_m) - t) * (y * -4.0)));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (x <= 1.05e+170)
		tmp = fma(x, x, Float64(Float64(Float64(z_m * z_m) - t) * Float64(y * -4.0)));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[x, 1.05e+170], N[(x * x + N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04999999999999999e170

   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 94.9%

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

      2. distribute-lft-neg-in 94.9%

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

      3. *-commutative 94.9%

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

      4. distribute-rgt-neg-in 94.9%

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

      5. metadata-eval 94.9%

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

   3. Simplified 94.9%

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

   if 1.04999999999999999e170 < x

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 93.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+298}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 2e+298)
   (+ (* x x) (* (* y 4.0) (- t (* z_m z_m))))
   (* 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 * z_m) <= 2e+298) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} 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 * z_m) <= 2d+298) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z_m * z_m)))
    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 * z_m) <= 2e+298) {
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	} 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 * z_m) <= 2e+298:
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)))
	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 (Float64(z_m * z_m) <= 2e+298)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z_m * z_m))));
	else
		tmp = Float64(z_m * Float64(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 * z_m) <= 2e+298)
		tmp = (x * x) + ((y * 4.0) * (t - (z_m * z_m)));
	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[N[(z$95$m * z$95$m), $MachinePrecision], 2e+298], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e298

   1. Initial program 96.8%

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

   if 1.9999999999999999e298 < (*.f64 z z)

   1. Initial program 76.0%

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

   3. Taylor expanded in z around inf 85.4%

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

      1. associate-*r* 85.4%

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

      2. *-commutative 85.4%

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

   5. Simplified 85.4%

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

      1. unpow2 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. add-sqr-sqrt 42.8%

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

      2. pow2 42.8%

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

      3. sqrt-prod 42.8%

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

      4. sqrt-prod 18.9%

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

      5. add-sqr-sqrt 47.0%

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

      6. *-commutative 47.0%

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

   9. Applied egg-rr 47.0%

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

       1. *-commutative 47.0%

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

       2. unpow-prod-down 42.8%

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

       3. pow2 42.8%

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

       4. add-sqr-sqrt 85.4%

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

       5. *-commutative 85.4%

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

       6. pow2 85.4%

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

       7. associate-*r* 93.9%

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

   11. Applied egg-rr 93.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+298}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+231}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 2e+231)
   (- (* x x) (* -4.0 (* y t)))
   (* 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 * z_m) <= 2e+231) {
		tmp = (x * x) - (-4.0 * (y * t));
	} 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 * z_m) <= 2d+231) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    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 * z_m) <= 2e+231) {
		tmp = (x * x) - (-4.0 * (y * t));
	} 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 * z_m) <= 2e+231:
		tmp = (x * x) - (-4.0 * (y * t))
	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 (Float64(z_m * z_m) <= 2e+231)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(z_m * Float64(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 * z_m) <= 2e+231)
		tmp = (x * x) - (-4.0 * (y * t));
	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[N[(z$95$m * z$95$m), $MachinePrecision], 2e+231], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e231

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 84.8%

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

      1. *-commutative 84.8%

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

   5. Simplified 84.8%

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

   if 2.0000000000000001e231 < (*.f64 z z)

   1. Initial program 79.8%

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

   3. Taylor expanded in z around inf 83.9%

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

      1. associate-*r* 83.9%

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

      2. *-commutative 83.9%

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

   5. Simplified 83.9%

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

      1. unpow2 83.9%

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

   7. Applied egg-rr 83.9%

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

      1. add-sqr-sqrt 37.4%

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

      2. pow2 37.4%

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

      3. sqrt-prod 37.4%

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

      4. sqrt-prod 15.9%

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

      5. add-sqr-sqrt 41.0%

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

      6. *-commutative 41.0%

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

   9. Applied egg-rr 41.0%

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

       1. *-commutative 41.0%

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

       2. unpow-prod-down 37.4%

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

       3. pow2 37.4%

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

       4. add-sqr-sqrt 83.9%

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

       5. *-commutative 83.9%

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

       6. pow2 83.9%

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

       7. associate-*r* 90.9%

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

   11. Applied egg-rr 90.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+231}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 8.5e+76) (* (* y 4.0) t) (* (* 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 * z_m) <= 8.5e+76) {
		tmp = (y * 4.0) * t;
	} 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 * z_m) <= 8.5d+76) then
        tmp = (y * 4.0d0) * t
    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 * z_m) <= 8.5e+76) {
		tmp = (y * 4.0) * t;
	} 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 * z_m) <= 8.5e+76:
		tmp = (y * 4.0) * t
	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 (Float64(z_m * z_m) <= 8.5e+76)
		tmp = Float64(Float64(y * 4.0) * t);
	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 * z_m) <= 8.5e+76)
		tmp = (y * 4.0) * t;
	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[N[(z$95$m * z$95$m), $MachinePrecision], 8.5e+76], N[(N[(y * 4.0), $MachinePrecision] * t), $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 (*.f64 z z) < 8.49999999999999992e76

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   5. Simplified 51.1%

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

   6. Taylor expanded in y around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

      3. *-commutative 51.1%

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

      4. associate-*r* 51.1%

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

   8. Simplified 51.1%

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

   if 8.49999999999999992e76 < (*.f64 z z)

   1. Initial program 83.3%

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

   3. Taylor expanded in z around inf 71.1%

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

      1. associate-*r* 71.1%

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

      2. *-commutative 71.1%

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

   5. Simplified 71.1%

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

      1. unpow2 71.1%

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

   7. Applied egg-rr 71.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 4 \cdot 10^{+61}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 4e+61) (* (* y 4.0) t) (* 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 * z_m) <= 4e+61) {
		tmp = (y * 4.0) * t;
	} 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 * z_m) <= 4d+61) then
        tmp = (y * 4.0d0) * t
    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 * z_m) <= 4e+61) {
		tmp = (y * 4.0) * t;
	} 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 * z_m) <= 4e+61:
		tmp = (y * 4.0) * t
	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 (Float64(z_m * z_m) <= 4e+61)
		tmp = Float64(Float64(y * 4.0) * t);
	else
		tmp = Float64(z_m * Float64(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 * z_m) <= 4e+61)
		tmp = (y * 4.0) * t;
	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[N[(z$95$m * z$95$m), $MachinePrecision], 4e+61], N[(N[(y * 4.0), $MachinePrecision] * t), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e61

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   5. Simplified 51.1%

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

   6. Taylor expanded in y around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

      3. *-commutative 51.1%

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

      4. associate-*r* 51.1%

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

   8. Simplified 51.1%

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

   if 3.9999999999999998e61 < (*.f64 z z)

   1. Initial program 83.3%

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

   3. Taylor expanded in z around inf 71.1%

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

      1. associate-*r* 71.1%

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

      2. *-commutative 71.1%

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

   5. Simplified 71.1%

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

      1. unpow2 71.1%

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

   7. Applied egg-rr 71.1%

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

      1. add-sqr-sqrt 32.6%

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

      2. pow2 32.6%

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

      3. sqrt-prod 32.6%

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

      4. sqrt-prod 14.7%

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

      5. add-sqr-sqrt 35.1%

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

      6. *-commutative 35.1%

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

   9. Applied egg-rr 35.1%

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

       1. *-commutative 35.1%

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

       2. unpow-prod-down 32.6%

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

       3. pow2 32.6%

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

       4. add-sqr-sqrt 71.1%

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

       5. *-commutative 71.1%

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

       6. pow2 71.1%

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

       7. associate-*r* 76.0%

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

   11. Applied egg-rr 76.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+61}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.6e+151) (* (* y 4.0) t) (* -4.0 (* y t))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+151) {
		tmp = (y * 4.0) * t;
	} else {
		tmp = -4.0 * (y * t);
	}
	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 <= 3.6d+151) then
        tmp = (y * 4.0d0) * t
    else
        tmp = (-4.0d0) * (y * t)
    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 <= 3.6e+151) {
		tmp = (y * 4.0) * t;
	} else {
		tmp = -4.0 * (y * t);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 3.6e+151:
		tmp = (y * 4.0) * t
	else:
		tmp = -4.0 * (y * t)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.6e+151)
		tmp = Float64(Float64(y * 4.0) * t);
	else
		tmp = Float64(-4.0 * Float64(y * t));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 3.6e+151)
		tmp = (y * 4.0) * t;
	else
		tmp = -4.0 * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.6e+151], N[(N[(y * 4.0), $MachinePrecision] * t), $MachinePrecision], N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.6e151

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

   5. Simplified 38.5%

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

   6. Taylor expanded in y around 0 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

      3. *-commutative 38.5%

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

      4. associate-*r* 38.9%

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

   8. Simplified 38.9%

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

   if 3.6e151 < z

   1. Initial program 74.6%

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

   3. Taylor expanded in z around 0 16.3%

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

      1. *-commutative 16.3%

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

   5. Simplified 16.3%

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

      1. associate-*r* 16.3%

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

      2. add-sqr-sqrt 12.6%

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

      3. sqrt-unprod 36.0%

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

      4. *-commutative 36.0%

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

      5. *-commutative 36.0%

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

      6. swap-sqr 36.0%

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

      7. metadata-eval 36.0%

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

      8. metadata-eval 36.0%

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

      9. swap-sqr 36.0%

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

      10. sqrt-unprod 15.2%

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

      11. add-sqr-sqrt 30.4%

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

      12. *-commutative 30.4%

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

      13. associate-*r* 30.4%

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

      14. metadata-eval 30.4%

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

      15. distribute-lft-neg-in 30.4%

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

      16. associate-*r* 30.4%

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

      17. *-commutative 30.4%

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

      18. associate-*l* 30.4%

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

   7. Applied egg-rr 30.4%

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

   8. Taylor expanded in x around 0 16.2%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 5.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 = (-4.0d0) * (y * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in z around 0 66.8%

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

   1. *-commutative 66.8%

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

5. Simplified 66.8%

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

   1. associate-*r* 67.1%

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

   2. add-sqr-sqrt 34.9%

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

   3. sqrt-unprod 47.7%

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

   4. *-commutative 47.7%

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

   5. *-commutative 47.7%

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

   6. swap-sqr 47.7%

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

   7. metadata-eval 47.7%

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

   8. metadata-eval 47.7%

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

   9. swap-sqr 47.7%

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

   10. sqrt-unprod 20.3%

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

   11. add-sqr-sqrt 40.0%

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

   12. *-commutative 40.0%

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

   13. associate-*r* 40.0%

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

   14. metadata-eval 40.0%

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

   15. distribute-lft-neg-in 40.0%

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

   16. associate-*r* 40.0%

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

   17. *-commutative 40.0%

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

   18. associate-*l* 40.0%

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

7. Applied egg-rr 40.0%

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

8. Taylor expanded in x around 0 7.6%

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

9. Final simplification 7.6%

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

Developer target: 91.5% 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 2024044 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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