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

Percentage Accurate: 90.6% → 95.5%

Time: 8.1s

Alternatives: 8

Speedup: 0.7×

• 50.3% 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: 90.6% 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: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t_1 \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(z \cdot -4\right) + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 4e+291)
     (fma x x (* t_1 (* y -4.0)))
     (+ (* (* z y) (* z -4.0)) (* x x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= 4e+291) {
		tmp = fma(x, x, (t_1 * (y * -4.0)));
	} else {
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= 4e+291)
		tmp = fma(x, x, Float64(t_1 * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(Float64(z * y) * Float64(z * -4.0)) + Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+291], N[(x * x + N[(t$95$1 * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 z z) t) < 3.9999999999999998e291

   1. Initial program 98.3%

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

      1. fma-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   if 3.9999999999999998e291 < (-.f64 (*.f64 z z) t)

   1. Initial program 71.7%

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

   2. Taylor expanded in t around 0 73.0%

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

      1. unpow2 73.0%

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

      2. fma-neg 81.4%

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

      3. distribute-lft-neg-in 81.4%

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

      4. metadata-eval 81.4%

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

      5. *-commutative 81.4%

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

      6. unpow2 81.4%

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

      7. associate-*l* 81.4%

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

   4. Simplified 81.4%

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

      1. fma-udef 73.0%

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

      2. +-commutative 73.0%

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

      3. associate-*l* 73.0%

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

      4. associate-*r* 88.8%

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

   6. Applied egg-rr 88.8%

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

4. Final simplification 96.5%

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

Alternative 2: 49.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t \cdot 4\right)\\ t_2 := z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-26} \lor \neg \left(x \leq 2.2 \cdot 10^{+60}\right) \land x \leq 5.4 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* t 4.0))) (t_2 (* z (* y (* z -4.0)))))
   (if (<= x 4.5e-248)
     t_2
     (if (<= x 6.4e-146)
       t_1
       (if (<= x 2e-131)
         t_2
         (if (<= x 4.7e-104)
           t_1
           (if (or (<= x 9.8e-26) (and (not (<= x 2.2e+60)) (<= x 5.4e+107)))
             t_2
             (* x x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t * 4.0);
	double t_2 = z * (y * (z * -4.0));
	double tmp;
	if (x <= 4.5e-248) {
		tmp = t_2;
	} else if (x <= 6.4e-146) {
		tmp = t_1;
	} else if (x <= 2e-131) {
		tmp = t_2;
	} else if (x <= 4.7e-104) {
		tmp = t_1;
	} else if ((x <= 9.8e-26) || (!(x <= 2.2e+60) && (x <= 5.4e+107))) {
		tmp = t_2;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * (t * 4.0d0)
    t_2 = z * (y * (z * (-4.0d0)))
    if (x <= 4.5d-248) then
        tmp = t_2
    else if (x <= 6.4d-146) then
        tmp = t_1
    else if (x <= 2d-131) then
        tmp = t_2
    else if (x <= 4.7d-104) then
        tmp = t_1
    else if ((x <= 9.8d-26) .or. (.not. (x <= 2.2d+60)) .and. (x <= 5.4d+107)) then
        tmp = t_2
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (t * 4.0);
	double t_2 = z * (y * (z * -4.0));
	double tmp;
	if (x <= 4.5e-248) {
		tmp = t_2;
	} else if (x <= 6.4e-146) {
		tmp = t_1;
	} else if (x <= 2e-131) {
		tmp = t_2;
	} else if (x <= 4.7e-104) {
		tmp = t_1;
	} else if ((x <= 9.8e-26) || (!(x <= 2.2e+60) && (x <= 5.4e+107))) {
		tmp = t_2;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (t * 4.0)
	t_2 = z * (y * (z * -4.0))
	tmp = 0
	if x <= 4.5e-248:
		tmp = t_2
	elif x <= 6.4e-146:
		tmp = t_1
	elif x <= 2e-131:
		tmp = t_2
	elif x <= 4.7e-104:
		tmp = t_1
	elif (x <= 9.8e-26) or (not (x <= 2.2e+60) and (x <= 5.4e+107)):
		tmp = t_2
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t * 4.0))
	t_2 = Float64(z * Float64(y * Float64(z * -4.0)))
	tmp = 0.0
	if (x <= 4.5e-248)
		tmp = t_2;
	elseif (x <= 6.4e-146)
		tmp = t_1;
	elseif (x <= 2e-131)
		tmp = t_2;
	elseif (x <= 4.7e-104)
		tmp = t_1;
	elseif ((x <= 9.8e-26) || (!(x <= 2.2e+60) && (x <= 5.4e+107)))
		tmp = t_2;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (t * 4.0);
	t_2 = z * (y * (z * -4.0));
	tmp = 0.0;
	if (x <= 4.5e-248)
		tmp = t_2;
	elseif (x <= 6.4e-146)
		tmp = t_1;
	elseif (x <= 2e-131)
		tmp = t_2;
	elseif (x <= 4.7e-104)
		tmp = t_1;
	elseif ((x <= 9.8e-26) || (~((x <= 2.2e+60)) && (x <= 5.4e+107)))
		tmp = t_2;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.5e-248], t$95$2, If[LessEqual[x, 6.4e-146], t$95$1, If[LessEqual[x, 2e-131], t$95$2, If[LessEqual[x, 4.7e-104], t$95$1, If[Or[LessEqual[x, 9.8e-26], And[N[Not[LessEqual[x, 2.2e+60]], $MachinePrecision], LessEqual[x, 5.4e+107]]], t$95$2, N[(x * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.4999999999999996e-248 or 6.3999999999999998e-146 < x < 2e-131 or 4.7e-104 < x < 9.7999999999999998e-26 or 2.19999999999999996e60 < x < 5.4000000000000003e107

   1. Initial program 93.2%

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

   2. Taylor expanded in z around inf 43.7%

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

      1. *-commutative 43.7%

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

      2. unpow2 43.7%

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

      3. associate-*l* 43.7%

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

   4. Simplified 43.7%

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

   5. Taylor expanded in y around 0 43.7%

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

      1. *-commutative 43.7%

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

      2. unpow2 43.7%

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

      3. associate-*r* 45.1%

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

      4. associate-*r* 45.1%

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

      5. *-commutative 45.1%

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

      6. associate-*l* 45.1%

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

      7. *-commutative 45.1%

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

   7. Simplified 45.1%

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

   if 4.4999999999999996e-248 < x < 6.3999999999999998e-146 or 2e-131 < x < 4.7e-104

   1. Initial program 91.1%

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

   2. Taylor expanded in t around inf 67.7%

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

      1. associate-*r* 67.7%

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

      2. *-commutative 67.7%

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

   4. Simplified 67.7%

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

   if 9.7999999999999998e-26 < x < 2.19999999999999996e60 or 5.4000000000000003e107 < x

   1. Initial program 84.5%

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

   2. Taylor expanded in x around inf 75.5%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-248}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-131}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-26} \lor \neg \left(x \leq 2.2 \cdot 10^{+60}\right) \land x \leq 5.4 \cdot 10^{+107}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-46} \lor \neg \left(x \cdot x \leq 2.2 \cdot 10^{+110}\right) \land x \cdot x \leq 7.5 \cdot 10^{+216}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x x) 1.8e-46)
         (and (not (<= (* x x) 2.2e+110)) (<= (* x x) 7.5e+216)))
   (* (- (* z z) t) (* y -4.0))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 1.8e-46) || (!((x * x) <= 2.2e+110) && ((x * x) <= 7.5e+216))) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) <= 1.8d-46) .or. (.not. ((x * x) <= 2.2d+110)) .and. ((x * x) <= 7.5d+216)) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 1.8e-46) || (!((x * x) <= 2.2e+110) && ((x * x) <= 7.5e+216))) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) <= 1.8e-46) or (not ((x * x) <= 2.2e+110) and ((x * x) <= 7.5e+216)):
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * x) <= 1.8e-46) || (!(Float64(x * x) <= 2.2e+110) && (Float64(x * x) <= 7.5e+216)))
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) <= 1.8e-46) || (~(((x * x) <= 2.2e+110)) && ((x * x) <= 7.5e+216)))
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.8e-46], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 2.2e+110]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 7.5e+216]]], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.8e-46 or 2.19999999999999992e110 < (*.f64 x x) < 7.4999999999999994e216

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 92.1%

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

      1. associate-*r* 92.1%

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

      2. unpow2 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

   4. Simplified 92.1%

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

   if 1.8e-46 < (*.f64 x x) < 2.19999999999999992e110 or 7.4999999999999994e216 < (*.f64 x x)

   1. Initial program 84.6%

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

   2. Taylor expanded in x around inf 78.6%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 78.6%

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

   4. Simplified 78.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-46} \lor \neg \left(x \cdot x \leq 2.2 \cdot 10^{+110}\right) \land x \cdot x \leq 7.5 \cdot 10^{+216}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ t_2 := y \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;x \leq 6 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-25} \lor \neg \left(x \leq 2 \cdot 10^{+61}\right) \land x \leq 4.5 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* (* z z) -4.0))) (t_2 (* y (* t 4.0))))
   (if (<= x 6e-147)
     t_2
     (if (<= x 1.65e-131)
       t_1
       (if (<= x 3.6e-103)
         t_2
         (if (or (<= x 3.2e-25) (and (not (<= x 2e+61)) (<= x 4.5e+107)))
           t_1
           (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z * z) * -4.0);
	double t_2 = y * (t * 4.0);
	double tmp;
	if (x <= 6e-147) {
		tmp = t_2;
	} else if (x <= 1.65e-131) {
		tmp = t_1;
	} else if (x <= 3.6e-103) {
		tmp = t_2;
	} else if ((x <= 3.2e-25) || (!(x <= 2e+61) && (x <= 4.5e+107))) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * ((z * z) * (-4.0d0))
    t_2 = y * (t * 4.0d0)
    if (x <= 6d-147) then
        tmp = t_2
    else if (x <= 1.65d-131) then
        tmp = t_1
    else if (x <= 3.6d-103) then
        tmp = t_2
    else if ((x <= 3.2d-25) .or. (.not. (x <= 2d+61)) .and. (x <= 4.5d+107)) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * ((z * z) * -4.0);
	double t_2 = y * (t * 4.0);
	double tmp;
	if (x <= 6e-147) {
		tmp = t_2;
	} else if (x <= 1.65e-131) {
		tmp = t_1;
	} else if (x <= 3.6e-103) {
		tmp = t_2;
	} else if ((x <= 3.2e-25) || (!(x <= 2e+61) && (x <= 4.5e+107))) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * ((z * z) * -4.0)
	t_2 = y * (t * 4.0)
	tmp = 0
	if x <= 6e-147:
		tmp = t_2
	elif x <= 1.65e-131:
		tmp = t_1
	elif x <= 3.6e-103:
		tmp = t_2
	elif (x <= 3.2e-25) or (not (x <= 2e+61) and (x <= 4.5e+107)):
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z * z) * -4.0))
	t_2 = Float64(y * Float64(t * 4.0))
	tmp = 0.0
	if (x <= 6e-147)
		tmp = t_2;
	elseif (x <= 1.65e-131)
		tmp = t_1;
	elseif (x <= 3.6e-103)
		tmp = t_2;
	elseif ((x <= 3.2e-25) || (!(x <= 2e+61) && (x <= 4.5e+107)))
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * ((z * z) * -4.0);
	t_2 = y * (t * 4.0);
	tmp = 0.0;
	if (x <= 6e-147)
		tmp = t_2;
	elseif (x <= 1.65e-131)
		tmp = t_1;
	elseif (x <= 3.6e-103)
		tmp = t_2;
	elseif ((x <= 3.2e-25) || (~((x <= 2e+61)) && (x <= 4.5e+107)))
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6e-147], t$95$2, If[LessEqual[x, 1.65e-131], t$95$1, If[LessEqual[x, 3.6e-103], t$95$2, If[Or[LessEqual[x, 3.2e-25], And[N[Not[LessEqual[x, 2e+61]], $MachinePrecision], LessEqual[x, 4.5e+107]]], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.0000000000000004e-147 or 1.6500000000000001e-131 < x < 3.5999999999999998e-103

   1. Initial program 92.0%

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

   2. Taylor expanded in t around inf 40.6%

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

      1. associate-*r* 40.6%

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

      2. *-commutative 40.6%

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

   4. Simplified 40.6%

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

   if 6.0000000000000004e-147 < x < 1.6500000000000001e-131 or 3.5999999999999998e-103 < x < 3.2000000000000001e-25 or 1.9999999999999999e61 < x < 4.5e107

   1. Initial program 97.3%

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

   2. Taylor expanded in z around inf 67.9%

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

      1. *-commutative 67.9%

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

      2. unpow2 67.9%

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

      3. associate-*l* 67.9%

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

   4. Simplified 67.9%

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

   if 3.2000000000000001e-25 < x < 1.9999999999999999e61 or 4.5e107 < x

   1. Initial program 84.5%

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

   2. Taylor expanded in x around inf 75.5%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-25} \lor \neg \left(x \leq 2 \cdot 10^{+61}\right) \land x \leq 4.5 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 95.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* z z) t) 4e+291)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* (* z y) (* z -4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) - t) <= 4e+291) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) - t) <= 4d+291) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = ((z * y) * (z * (-4.0d0))) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) - t) <= 4e+291) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) - t) <= 4e+291:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = ((z * y) * (z * -4.0)) + (x * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) - t) <= 4e+291)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(z * y) * Float64(z * -4.0)) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) - t) <= 4e+291)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision], 4e+291], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 z z) t) < 3.9999999999999998e291

   1. Initial program 98.3%

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

   if 3.9999999999999998e291 < (-.f64 (*.f64 z z) t)

   1. Initial program 71.7%

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

   2. Taylor expanded in t around 0 73.0%

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

      1. unpow2 73.0%

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

      2. fma-neg 81.4%

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

      3. distribute-lft-neg-in 81.4%

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

      4. metadata-eval 81.4%

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

      5. *-commutative 81.4%

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

      6. unpow2 81.4%

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

      7. associate-*l* 81.4%

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

   4. Simplified 81.4%

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

      1. fma-udef 73.0%

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

      2. +-commutative 73.0%

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

      3. associate-*l* 73.0%

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

      4. associate-*r* 88.8%

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

   6. Applied egg-rr 88.8%

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

4. Final simplification 95.7%

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

Alternative 6: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-46}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(z \cdot -4\right) + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e-46)
   (* (- (* z z) t) (* y -4.0))
   (+ (* (* z y) (* z -4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-46) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1d-46) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = ((z * y) * (z * (-4.0d0))) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-46) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1e-46:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = ((z * y) * (z * -4.0)) + (x * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1e-46)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(Float64(Float64(z * y) * Float64(z * -4.0)) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1e-46)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = ((z * y) * (z * -4.0)) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-46], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000002e-46

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 96.6%

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

      1. associate-*r* 96.6%

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

      2. unpow2 96.6%

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

      3. *-commutative 96.6%

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

      4. *-commutative 96.6%

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

   4. Simplified 96.6%

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

   if 1.00000000000000002e-46 < (*.f64 x x)

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0 78.8%

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

      1. unpow2 78.8%

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

      2. fma-neg 83.6%

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

      3. distribute-lft-neg-in 83.6%

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

      4. metadata-eval 83.6%

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

      5. *-commutative 83.6%

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

      6. unpow2 83.6%

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

      7. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

      1. fma-udef 78.8%

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

      2. +-commutative 78.8%

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

      3. associate-*l* 78.8%

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

      4. associate-*r* 84.1%

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

   6. Applied egg-rr 84.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-46}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(z \cdot -4\right) + x \cdot x\\ \end{array} \]

Alternative 7: 59.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.8e-46) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.8e-46) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1.8d-46) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.8e-46) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.8e-46:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.8e-46)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.8e-46)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.8e-46], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.8e-46

   1. Initial program 96.6%

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

   2. Taylor expanded in t around inf 54.7%

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

      1. associate-*r* 54.7%

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

      2. *-commutative 54.7%

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

   4. Simplified 54.7%

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

   if 1.8e-46 < (*.f64 x x)

   1. Initial program 86.5%

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

   2. Taylor expanded in x around inf 70.6%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 70.6%

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

   4. Simplified 70.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 41.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

2. Taylor expanded in x around inf 42.8%

   \[\leadsto \color{blue}{{x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 42.8%

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

4. Simplified 42.8%

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

5. Final simplification 42.8%

   \[\leadsto x \cdot x \]

Developer target: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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))))