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

Percentage Accurate: 90.5% → 96.0%

Time: 11.7s

Alternatives: 8

Speedup: 0.8×

• 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: 90.5% 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: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+252}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+252)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+252) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 2d+252) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - ((z * y) * (z * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+252) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+252:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - ((z * y) * (z * 4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+252)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+252)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - ((z * y) * (z * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+252], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e252

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   if 2.0000000000000002e252 < (*.f64 z z)

   1. Initial program 79.9%

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

      1. flip3-- 0.0%

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

      2. associate-*r/ 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. swap-sqr 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto x \cdot x - \frac{\sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left({\left(z \cdot z\right)}^{3} - {t}^{3}\right)}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(t \cdot t + \left(z \cdot z\right) \cdot t\right)} \]

      7. metadata-eval 0.0%

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

      8. swap-sqr 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. associate-/l* 0.0%

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

   3. Applied egg-rr 79.9%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{{z}^{2} - t}}} \]

   4. Taylor expanded in z around inf 79.9%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. div-inv 79.9%

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

      2. remove-double-div 79.9%

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

      3. unpow2 79.9%

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

      4. associate-*l* 90.3%

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

      5. *-commutative 90.3%

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

      6. associate-*l* 90.3%

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

      7. associate-*r* 90.3%

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

   6. Applied egg-rr 90.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+252}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]

Alternative 2: 92.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. fma-neg 95.4%

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

   2. *-commutative 95.4%

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

   3. distribute-rgt-neg-in 95.4%

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

   4. distribute-lft-neg-out 95.4%

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

   5. distribute-lft-neg-out 95.4%

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

   6. distribute-rgt-neg-in 95.4%

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

   7. metadata-eval 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 3: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot \left(z \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 8e-37)
   (- (* x x) (* -4.0 (* t y)))
   (- (* x x) (* y (* z (* z 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8e-37) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (y * (z * (z * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 8d-37) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = (x * x) - (y * (z * (z * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8e-37) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (y * (z * (z * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 8e-37:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = (x * x) - (y * (z * (z * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8e-37)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(z * Float64(z * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8e-37)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = (x * x) - (y * (z * (z * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 8e-37], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(y * N[(z * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.00000000000000053e-37

   1. Initial program 94.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 70.6%

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

      1. *-commutative 70.6%

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

   4. Simplified 70.6%

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

   if 8.00000000000000053e-37 < z

   1. Initial program 89.7%

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

      1. flip3-- 25.1%

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

      2. associate-*r/ 21.3%

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

      3. add-sqr-sqrt 6.7%

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

      4. sqrt-unprod 10.9%

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

      5. swap-sqr 10.9%

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

      6. metadata-eval 10.9%

         \[\leadsto x \cdot x - \frac{\sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left({\left(z \cdot z\right)}^{3} - {t}^{3}\right)}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(t \cdot t + \left(z \cdot z\right) \cdot t\right)} \]

      7. metadata-eval 10.9%

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

      8. swap-sqr 10.9%

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

      9. sqrt-unprod 6.7%

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

      10. add-sqr-sqrt 10.7%

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

      11. associate-/l* 10.7%

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

   3. Applied egg-rr 89.7%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{{z}^{2} - t}}} \]

   4. Taylor expanded in z around inf 85.6%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. div-inv 85.5%

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

      2. remove-double-div 85.6%

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

      3. unpow2 85.6%

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

      4. associate-*l* 93.1%

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

      5. associate-*l* 93.1%

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

      6. associate-*l* 85.6%

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

   6. Applied egg-rr 85.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot \left(z \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 8e-37)
   (- (* x x) (* -4.0 (* t y)))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8e-37) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 8d-37) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = (x * x) - ((z * y) * (z * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8e-37) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 8e-37:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = (x * x) - ((z * y) * (z * 4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8e-37)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8e-37)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = (x * x) - ((z * y) * (z * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 8e-37], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.00000000000000053e-37

   1. Initial program 94.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 70.6%

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

      1. *-commutative 70.6%

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

   4. Simplified 70.6%

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

   if 8.00000000000000053e-37 < z

   1. Initial program 89.7%

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

      1. flip3-- 25.1%

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

      2. associate-*r/ 21.3%

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

      3. add-sqr-sqrt 6.7%

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

      4. sqrt-unprod 10.9%

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

      5. swap-sqr 10.9%

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

      6. metadata-eval 10.9%

         \[\leadsto x \cdot x - \frac{\sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left({\left(z \cdot z\right)}^{3} - {t}^{3}\right)}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(t \cdot t + \left(z \cdot z\right) \cdot t\right)} \]

      7. metadata-eval 10.9%

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

      8. swap-sqr 10.9%

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

      9. sqrt-unprod 6.7%

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

      10. add-sqr-sqrt 10.7%

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

      11. associate-/l* 10.7%

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

   3. Applied egg-rr 89.7%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{{z}^{2} - t}}} \]

   4. Taylor expanded in z around inf 85.6%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. div-inv 85.5%

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

      2. remove-double-div 85.6%

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

      3. unpow2 85.6%

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

      4. associate-*l* 93.1%

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

      5. *-commutative 93.1%

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

      6. associate-*l* 93.1%

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

      7. associate-*r* 93.1%

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

   6. Applied egg-rr 93.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-37}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]

Alternative 5: 65.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.9e+102) (- (* x x) (* -4.0 (* t y))) (- (* x x) (* 4.0 (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+102) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (4.0 * (t * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.9d+102) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = (x * x) - (4.0d0 * (t * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+102) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (4.0 * (t * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.9e+102:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = (x * x) - (4.0 * (t * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.9e+102)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(4.0 * Float64(t * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.9e+102)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = (x * x) - (4.0 * (t * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.9e+102], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.89999999999999989e102

   1. Initial program 95.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 67.5%

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

      1. *-commutative 67.5%

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

   4. Simplified 67.5%

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

   if 1.89999999999999989e102 < z

   1. Initial program 82.7%

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

      1. flip3-- 0.0%

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

      2. associate-*r/ 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. swap-sqr 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto x \cdot x - \frac{\sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left({\left(z \cdot z\right)}^{3} - {t}^{3}\right)}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(t \cdot t + \left(z \cdot z\right) \cdot t\right)} \]

      7. metadata-eval 0.0%

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

      8. swap-sqr 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. associate-/l* 0.0%

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

   3. Applied egg-rr 82.8%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{{z}^{2} - t}}} \]

   4. Taylor expanded in z around 0 16.5%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{-1}{t}}} \]
   5. Step-by-step derivation

      1. clear-num 16.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{1}{\frac{\frac{-1}{t}}{y \cdot 4}}} \]

      2. metadata-eval 16.5%

         \[\leadsto x \cdot x - \frac{\color{blue}{\frac{2}{2}}}{\frac{\frac{-1}{t}}{y \cdot 4}} \]

      3. associate-/r/ 16.5%

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

      4. associate-/r/ 16.5%

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

      5. metadata-eval 16.5%

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

      6. metadata-eval 16.5%

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

      7. neg-mul-1 16.5%

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

      8. add-sqr-sqrt 9.1%

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

      9. sqrt-unprod 25.0%

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

      10. sqr-neg 25.0%

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

      11. sqrt-unprod 12.6%

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

      12. add-sqr-sqrt 15.5%

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

      13. associate-*l* 15.5%

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

      14. *-commutative 15.5%

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

   6. Applied egg-rr 15.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 6: 66.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in z around 0 58.5%

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

   1. *-commutative 58.5%

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

4. Simplified 58.5%

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

5. Final simplification 58.5%

   \[\leadsto x \cdot x - -4 \cdot \left(t \cdot y\right) \]

Alternative 7: 31.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.9e+102) (* y (* t 4.0)) (* y (* t -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+102) {
		tmp = y * (t * 4.0);
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.9d+102) then
        tmp = y * (t * 4.0d0)
    else
        tmp = y * (t * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+102) {
		tmp = y * (t * 4.0);
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.9e+102:
		tmp = y * (t * 4.0)
	else:
		tmp = y * (t * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.9e+102)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(y * Float64(t * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.9e+102)
		tmp = y * (t * 4.0);
	else
		tmp = y * (t * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.9e+102], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.89999999999999989e102

   1. Initial program 95.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf 32.7%

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

      1. associate-*r* 32.7%

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

      2. *-commutative 32.7%

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

      3. *-commutative 32.7%

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

   4. Simplified 32.7%

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

   if 1.89999999999999989e102 < z

   1. Initial program 82.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf 7.6%

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

      1. *-commutative 7.6%

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

      2. *-commutative 7.6%

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

   4. Simplified 7.6%

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

      1. *-commutative 7.6%

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

      2. associate-*l* 7.6%

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

      3. add-sqr-sqrt 0.6%

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

      4. sqrt-unprod 3.3%

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

      5. sqr-neg 3.3%

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

      6. sqrt-unprod 0.5%

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

      7. add-sqr-sqrt 6.5%

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

      8. neg-mul-1 6.5%

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

      9. metadata-eval 6.5%

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

      10. metadata-eval 6.5%

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

      11. associate-/r/ 6.5%

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

      12. associate-/r/ 6.5%

          \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\frac{\frac{-1}{t}}{y \cdot 4}}} \]

      13. metadata-eval 6.5%

          \[\leadsto \frac{\color{blue}{1}}{\frac{\frac{-1}{t}}{y \cdot 4}} \]

      14. clear-num 6.5%

          \[\leadsto \color{blue}{\frac{y \cdot 4}{\frac{-1}{t}}} \]

      15. div-inv 6.5%

          \[\leadsto \frac{y \cdot 4}{\color{blue}{-1 \cdot \frac{1}{t}}} \]

      16. associate-/r* 6.5%

          \[\leadsto \color{blue}{\frac{\frac{y \cdot 4}{-1}}{\frac{1}{t}}} \]

      17. associate-/l* 6.5%

          \[\leadsto \frac{\color{blue}{\frac{y}{\frac{-1}{4}}}}{\frac{1}{t}} \]

      18. metadata-eval 6.5%

          \[\leadsto \frac{\frac{y}{\color{blue}{-0.25}}}{\frac{1}{t}} \]

      19. metadata-eval 6.5%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{\color{blue}{--1}}{t}} \]

      20. add-sqr-sqrt 6.0%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{--1}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}} \]

      21. sqrt-unprod 18.4%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{--1}{\color{blue}{\sqrt{t \cdot t}}}} \]

      22. sqr-neg 18.4%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{--1}{\sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}}} \]

      23. sqrt-unprod 7.0%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{--1}{\color{blue}{\sqrt{-t} \cdot \sqrt{-t}}}} \]

      24. add-sqr-sqrt 7.6%

          \[\leadsto \frac{\frac{y}{-0.25}}{\frac{--1}{\color{blue}{-t}}} \]

      25. frac-2neg 7.6%

          \[\leadsto \frac{\frac{y}{-0.25}}{\color{blue}{\frac{-1}{t}}} \]

   6. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{-0.25}}{\frac{-1}{t}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 7.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{-0.25}}{-1} \cdot t} \]

      2. associate-/l/ 7.6%

         \[\leadsto \color{blue}{\frac{y}{-1 \cdot -0.25}} \cdot t \]

      3. metadata-eval 7.6%

         \[\leadsto \frac{y}{\color{blue}{0.25}} \cdot t \]

      4. metadata-eval 7.6%

         \[\leadsto \frac{y}{\color{blue}{\frac{1}{4}}} \cdot t \]

      5. associate-/l* 7.6%

         \[\leadsto \color{blue}{\frac{y \cdot 4}{1}} \cdot t \]

      6. /-rgt-identity 7.6%

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

      7. add-sqr-sqrt 4.5%

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

      8. sqrt-unprod 13.5%

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

      9. swap-sqr 13.5%

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

      10. metadata-eval 13.5%

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

      11. metadata-eval 13.5%

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

      12. swap-sqr 13.5%

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

      13. metadata-eval 13.5%

          \[\leadsto \sqrt{\left(y \cdot \color{blue}{\frac{1}{-0.25}}\right) \cdot \left(y \cdot -4\right)} \cdot t \]

      14. div-inv 13.5%

          \[\leadsto \sqrt{\color{blue}{\frac{y}{-0.25}} \cdot \left(y \cdot -4\right)} \cdot t \]

      15. metadata-eval 13.5%

          \[\leadsto \sqrt{\frac{y}{-0.25} \cdot \left(y \cdot \color{blue}{\frac{1}{-0.25}}\right)} \cdot t \]

      16. div-inv 13.5%

          \[\leadsto \sqrt{\frac{y}{-0.25} \cdot \color{blue}{\frac{y}{-0.25}}} \cdot t \]

      17. sqrt-unprod 2.8%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{y}{-0.25}} \cdot \sqrt{\frac{y}{-0.25}}\right)} \cdot t \]

      18. add-sqr-sqrt 6.5%

          \[\leadsto \color{blue}{\frac{y}{-0.25}} \cdot t \]

      19. div-inv 6.5%

          \[\leadsto \color{blue}{\left(y \cdot \frac{1}{-0.25}\right)} \cdot t \]

      20. metadata-eval 6.5%

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

      21. *-commutative 6.5%

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

      22. associate-*r* 6.5%

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

      23. *-commutative 6.5%

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

      24. associate-*r* 6.5%

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

      25. *-commutative 6.5%

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

   8. Applied egg-rr 6.5%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \]

Alternative 8: 31.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in t around inf 28.3%

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

   1. associate-*r* 28.3%

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

   2. *-commutative 28.3%

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

   3. *-commutative 28.3%

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

4. Simplified 28.3%

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

5. Final simplification 28.3%

   \[\leadsto y \cdot \left(t \cdot 4\right) \]

Developer target: 90.5% 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 2023301 
(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))))