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

Percentage Accurate: 91.1% → 94.1%

Time: 12.1s

Alternatives: 8

Speedup: 0.4×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 94.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 4e+224)
   (fma x_m x_m (* (- (* z z) t) (* y -4.0)))
   (fma (* y 4.0) t (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 4e+224) {
		tmp = fma(x_m, x_m, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma((y * 4.0), t, (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 4e+224)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(Float64(y * 4.0), t, Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 4e+224], N[(x$95$m * x$95$m + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 4.0), $MachinePrecision] * t + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999988e224

   1. Initial program 91.0%

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

      1. fmm-def 93.9%

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

      2. distribute-lft-neg-in 93.9%

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

      3. *-commutative 93.9%

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

      4. distribute-rgt-neg-in 93.9%

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

      5. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   if 3.99999999999999988e224 < x

   1. Initial program 64.7%

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

      1. cancel-sign-sub-inv 64.7%

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

      2. distribute-lft-neg-out 64.7%

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

      3. +-commutative 64.7%

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

      4. distribute-lft-neg-out 64.7%

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

      5. distribute-lft-neg-in 64.7%

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

      6. distribute-rgt-neg-in 64.7%

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

      7. fma-define 82.4%

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

      8. sub-neg 82.4%

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

      9. +-commutative 82.4%

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

      10. distribute-neg-in 82.4%

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

      11. remove-double-neg 82.4%

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

      12. sub-neg 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in t around inf 100.0%

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

Alternative 2: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{+150}:\\ \;\;\;\;x\_m \cdot x\_m + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= x_m 1.2e+150)
   (+ (* x_m x_m) (* t (* 4.0 (- y (/ (* z y) (/ t z))))))
   (fma (* y 4.0) t (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.2e+150) {
		tmp = (x_m * x_m) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	} else {
		tmp = fma((y * 4.0), t, (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 1.2e+150)
		tmp = Float64(Float64(x_m * x_m) + Float64(t * Float64(4.0 * Float64(y - Float64(Float64(z * y) / Float64(t / z))))));
	else
		tmp = fma(Float64(y * 4.0), t, Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[x$95$m, 1.2e+150], N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(t * N[(4.0 * N[(y - N[(N[(z * y), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 4.0), $MachinePrecision] * t + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000001e150

   1. Initial program 92.0%

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

   3. Taylor expanded in t around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. *-commutative 89.8%

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

      3. *-commutative 89.8%

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

      4. metadata-eval 89.8%

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

      5. distribute-rgt-neg-in 89.8%

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

      6. distribute-lft-neg-in 89.8%

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

      7. distribute-rgt-out 89.8%

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

      8. unsub-neg 89.8%

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

      9. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

      1. pow2 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. associate-*r/ 91.0%

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

      2. associate-*r* 92.7%

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

      3. clear-num 92.7%

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

      4. un-div-inv 92.7%

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

   9. Applied egg-rr 92.7%

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

   if 1.20000000000000001e150 < x

   1. Initial program 73.0%

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

      1. cancel-sign-sub-inv 73.0%

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

      2. distribute-lft-neg-out 73.0%

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

      3. +-commutative 73.0%

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

      4. distribute-lft-neg-out 73.0%

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

      5. distribute-lft-neg-in 73.0%

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

      6. distribute-rgt-neg-in 73.0%

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

      7. fma-define 81.1%

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

      8. sub-neg 81.1%

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

      9. +-commutative 81.1%

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

      10. distribute-neg-in 81.1%

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

      11. remove-double-neg 81.1%

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

      12. sub-neg 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in t around inf 89.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+150}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 87.6%

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

   3. Taylor expanded in t around inf 87.6%

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

      1. +-commutative 87.6%

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

      2. *-commutative 87.6%

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

      3. *-commutative 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-rgt-neg-in 87.6%

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

      6. distribute-lft-neg-in 87.6%

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

      7. distribute-rgt-out 87.6%

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

      8. unsub-neg 87.6%

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

      9. associate-/l* 87.6%

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

   5. Simplified 87.6%

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

      1. pow2 87.6%

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

   7. Applied egg-rr 87.6%

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

      1. associate-*r/ 95.0%

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

      2. associate-*r* 100.0%

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

      3. clear-num 100.0%

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

      4. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

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

   1. Initial program 98.5%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 25.0%

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

      3. swap-sqr 25.0%

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

      4. metadata-eval 25.0%

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

      5. metadata-eval 25.0%

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

      6. swap-sqr 25.0%

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

      7. sqrt-unprod 25.0%

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

      8. add-sqr-sqrt 65.0%

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

      9. metadata-eval 65.0%

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

      10. distribute-rgt-neg-in 65.0%

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

   4. Applied egg-rr 65.0%

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

4. Final simplification 96.1%

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

Alternative 4: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.8%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 25.0%

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

      3. swap-sqr 25.0%

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

      4. metadata-eval 25.0%

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

      5. metadata-eval 25.0%

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

      6. swap-sqr 25.0%

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

      7. sqrt-unprod 25.0%

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

      8. add-sqr-sqrt 65.0%

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

      9. metadata-eval 65.0%

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

      10. distribute-rgt-neg-in 65.0%

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

   4. Applied egg-rr 65.0%

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

4. Final simplification 94.3%

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

Alternative 5: 93.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.8%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 25.0%

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

      3. swap-sqr 25.0%

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

      4. metadata-eval 25.0%

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

      5. metadata-eval 25.0%

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

      6. swap-sqr 25.0%

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

      7. sqrt-unprod 25.0%

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

      8. add-sqr-sqrt 65.0%

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

      9. metadata-eval 65.0%

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

      10. distribute-rgt-neg-in 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in z around 0 60.0%

      \[\leadsto x \cdot x - \left(-y \cdot 4\right) \cdot \color{blue}{\left(-1 \cdot t\right)} \]
   6. Step-by-step derivation

      1. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in y around 0 60.0%

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

      1. *-commutative 60.0%

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

   10. Simplified 60.0%

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

4. Final simplification 93.9%

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

Alternative 6: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.6 \cdot 10^{+203}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 4.6e+203)
   (- (* x_m x_m) (* y (* t -4.0)))
   (* (* z z) (* y -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4.6e+203) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4.6d+203) then
        tmp = (x_m * x_m) - (y * (t * (-4.0d0)))
    else
        tmp = (z * z) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4.6e+203) {
		tmp = (x_m * x_m) - (y * (t * -4.0));
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 4.6e+203:
		tmp = (x_m * x_m) - (y * (t * -4.0))
	else:
		tmp = (z * z) * (y * -4.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4.6e+203)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4.6e+203)
		tmp = (x_m * x_m) - (y * (t * -4.0));
	else
		tmp = (z * z) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.6e+203], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.5999999999999998e203

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-*l* 82.4%

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

   5. Simplified 82.4%

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

   if 4.5999999999999998e203 < (*.f64 z z)

   1. Initial program 78.8%

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

      1. fmm-def 83.1%

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

      2. distribute-lft-neg-in 83.1%

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

      3. *-commutative 83.1%

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

      4. distribute-rgt-neg-in 83.1%

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

      5. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in z around inf 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. *-commutative 75.3%

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

   7. Simplified 75.3%

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

      1. pow2 76.6%

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

   9. Applied egg-rr 75.3%

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

Alternative 7: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9.2 \cdot 10^{-44}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z t)
 :precision binary64
 (if (<= (* z z) 9.2e-44) (* 4.0 (* t y)) (* (* z z) (* y -4.0))))

C

x_m = fabs(x);
double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 9.2e-44) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z, t)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 9.2d-44) then
        tmp = 4.0d0 * (t * y)
    else
        tmp = (z * z) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 9.2e-44) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z, t):
	tmp = 0
	if (z * z) <= 9.2e-44:
		tmp = 4.0 * (t * y)
	else:
		tmp = (z * z) * (y * -4.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 9.2e-44)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 9.2e-44)
		tmp = 4.0 * (t * y);
	else
		tmp = (z * z) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 9.2e-44], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.19999999999999992e-44

   1. Initial program 95.3%

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

      1. fmm-def 96.3%

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

      2. distribute-lft-neg-in 96.3%

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

      3. *-commutative 96.3%

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

      4. distribute-rgt-neg-in 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t around inf 46.6%

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

      1. *-commutative 46.6%

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

   7. Simplified 46.6%

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

   if 9.19999999999999992e-44 < (*.f64 z z)

   1. Initial program 84.7%

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

      1. fmm-def 88.8%

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

      2. distribute-lft-neg-in 88.8%

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

      3. *-commutative 88.8%

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

      4. distribute-rgt-neg-in 88.8%

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

      5. metadata-eval 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in z around inf 58.4%

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

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

   7. Simplified 58.4%

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

      1. pow2 79.5%

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

   9. Applied egg-rr 58.4%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9.2 \cdot 10^{-44}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_, t_] := N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. fmm-def 91.9%

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

   2. distribute-lft-neg-in 91.9%

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

   3. *-commutative 91.9%

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

   4. distribute-rgt-neg-in 91.9%

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

   5. metadata-eval 91.9%

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

3. Simplified 91.9%

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

5. Taylor expanded in t around inf 30.5%

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

   1. *-commutative 30.5%

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

7. Simplified 30.5%

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

8. Final simplification 30.5%

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

Developer Target 1: 91.2% 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 2024172 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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