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

Percentage Accurate: 90.5% → 94.0%

Time: 13.9s

Alternatives: 8

Speedup: 0.4×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+300)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* -4.0 (* t (* y (fma z (/ z t) -1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+300) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = -4.0 * (t * (y * fma(z, (z / t), -1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+300)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * fma(z, Float64(z / t), -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+300], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(z * N[(z / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e300

   1. Initial program 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. distribute-lft-neg-out 98.9%

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

      3. +-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. associate-*l* 98.9%

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

      7. distribute-rgt-neg-in 98.9%

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

      8. fma-define 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. remove-double-neg 100.0%

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

      13. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 1.0000000000000001e300 < (*.f64 z z)

   1. Initial program 64.2%

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

   3. Taylor expanded in t around inf 64.2%

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

   4. Taylor expanded in x around 0 78.9%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 78.9%

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

      2. metadata-eval 78.9%

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

      3. associate-*r* 71.0%

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

      4. *-commutative 71.0%

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

      5. metadata-eval 71.0%

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

      6. sub-neg 71.0%

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

      7. unpow2 71.0%

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

      8. associate-*r/ 75.1%

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

      9. fma-neg 75.1%

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

      10. metadata-eval 75.1%

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

   6. Simplified 75.1%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 71.0%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)} \]

      2. pow1 71.0%

         \[\leadsto -4 \cdot \color{blue}{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\right)}^{1}} \]

      3. log1p-expm1-u 75.1%

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

      4. *-commutative 75.1%

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

      5. metadata-eval 75.1%

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

      6. fma-neg 75.1%

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

      7. associate-*l* 83.0%

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

      8. fma-neg 83.0%

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

      9. metadata-eval 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto -4 \cdot \color{blue}{{\left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. pow1 83.0%

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

      2. pow1 83.0%

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

      3. associate-*r* 75.1%

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

   10. Applied egg-rr 75.1%

       \[\leadsto \color{blue}{{\left(-4 \cdot \left(\left(t \cdot y\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 75.1%

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

       2. associate-*l* 83.0%

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

   12. Simplified 83.0%

       \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-4 \cdot {z}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (* y (* -4.0 (pow z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = y * (-4.0 * pow(z, 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y * (-4.0 * Math.pow(z, 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y * (-4.0 * math.pow(z, 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(-4.0 * (z ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y * (-4.0 * (z ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $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[(y * N[(-4.0 * N[Power[z, 2.0], $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 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. Taylor expanded in z around inf 61.1%

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

      1. associate-*r* 61.1%

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

      2. *-commutative 61.1%

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

      3. associate-*l* 61.1%

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

   5. Simplified 61.1%

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

4. Final simplification 94.0%

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

Alternative 3: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+300)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* t (* y (fma z (/ z t) -1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+300) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (t * (y * fma(z, (z / t), -1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+300)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(t * Float64(y * fma(z, Float64(z / t), -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+300], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t * N[(y * N[(z * N[(z / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e300

   1. Initial program 98.9%

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

   if 1.0000000000000001e300 < (*.f64 z z)

   1. Initial program 64.2%

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

   3. Taylor expanded in t around inf 64.2%

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

   4. Taylor expanded in x around 0 78.9%

      \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \left(\frac{{z}^{2}}{t} - 1\right)\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 78.9%

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

      2. metadata-eval 78.9%

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

      3. associate-*r* 71.0%

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

      4. *-commutative 71.0%

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

      5. metadata-eval 71.0%

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

      6. sub-neg 71.0%

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

      7. unpow2 71.0%

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

      8. associate-*r/ 75.1%

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

      9. fma-neg 75.1%

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

      10. metadata-eval 75.1%

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

   6. Simplified 75.1%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 71.0%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)} \]

      2. pow1 71.0%

         \[\leadsto -4 \cdot \color{blue}{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(y \cdot t\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\right)}^{1}} \]

      3. log1p-expm1-u 75.1%

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

      4. *-commutative 75.1%

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

      5. metadata-eval 75.1%

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

      6. fma-neg 75.1%

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

      7. associate-*l* 83.0%

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

      8. fma-neg 83.0%

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

      9. metadata-eval 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto -4 \cdot \color{blue}{{\left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. pow1 83.0%

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

      2. pow1 83.0%

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

      3. associate-*r* 75.1%

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

   10. Applied egg-rr 75.1%

       \[\leadsto \color{blue}{{\left(-4 \cdot \left(\left(t \cdot y\right) \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 75.1%

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

       2. associate-*l* 83.0%

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

   12. Simplified 83.0%

       \[\leadsto \color{blue}{-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(y \cdot \mathsf{fma}\left(z, \frac{z}{t}, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

   1. fma-neg 93.6%

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

   2. distribute-lft-neg-in 93.6%

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

   3. *-commutative 93.6%

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

   4. distribute-rgt-neg-in 93.6%

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

   5. metadata-eval 93.6%

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

3. Simplified 93.6%

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

5. Final simplification 93.6%

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

Alternative 5: 93.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $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[(x * x), $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 96.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. sub-neg 0.0%

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

      2. distribute-lft-neg-in 0.0%

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

      3. *-commutative 0.0%

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

      4. distribute-lft-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 11.1%

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

      9. swap-sqr 11.1%

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

      10. metadata-eval 11.1%

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

      11. metadata-eval 11.1%

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

      12. swap-sqr 11.1%

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

      13. sqrt-unprod 38.9%

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

      14. add-sqr-sqrt 44.4%

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

      15. expm1-log1p-u 44.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      16. add-sqr-sqrt 38.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

   6. Simplified 44.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 44.4%

         \[\leadsto \color{blue}{{x}^{2}} \]

      2. pow2 44.4%

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

   8. Applied egg-rr 44.4%

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

4. Final simplification 92.8%

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

Alternative 6: 44.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-99} \lor \neg \left(x \leq 6.5 \cdot 10^{-72}\right) \land x \leq 7.2 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x 7e-99) (and (not (<= x 6.5e-72)) (<= x 7.2e+31)))
   (* y (* 4.0 t))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= 7e-99) || (!(x <= 6.5e-72) && (x <= 7.2e+31))) {
		tmp = y * (4.0 * t);
	} 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 <= 7d-99) .or. (.not. (x <= 6.5d-72)) .and. (x <= 7.2d+31)) then
        tmp = y * (4.0d0 * t)
    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 <= 7e-99) || (!(x <= 6.5e-72) && (x <= 7.2e+31))) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= 7e-99) or (not (x <= 6.5e-72) and (x <= 7.2e+31)):
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= 7e-99) || (!(x <= 6.5e-72) && (x <= 7.2e+31)))
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= 7e-99) || (~((x <= 6.5e-72)) && (x <= 7.2e+31)))
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, 7e-99], And[N[Not[LessEqual[x, 6.5e-72]], $MachinePrecision], LessEqual[x, 7.2e+31]]], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999997e-99 or 6.4999999999999997e-72 < x < 7.19999999999999992e31

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf 39.0%

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

      1. associate-*r* 39.0%

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

   5. Simplified 39.0%

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

   if 6.9999999999999997e-99 < x < 6.4999999999999997e-72 or 7.19999999999999992e31 < x

   1. Initial program 79.8%

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

      1. sub-neg 79.8%

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

      2. distribute-lft-neg-in 79.8%

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

      3. *-commutative 79.8%

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

      4. distribute-lft-neg-in 79.8%

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

      5. metadata-eval 79.8%

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

      6. *-commutative 79.8%

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

      7. add-sqr-sqrt 33.3%

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

      8. sqrt-unprod 65.2%

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

      9. swap-sqr 65.2%

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

      10. metadata-eval 65.2%

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

      11. metadata-eval 65.2%

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

      12. swap-sqr 65.2%

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

      13. sqrt-unprod 39.1%

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

      14. add-sqr-sqrt 62.3%

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

      15. expm1-log1p-u 60.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      16. add-sqr-sqrt 38.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

   6. Simplified 69.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 71.3%

         \[\leadsto \color{blue}{{x}^{2}} \]

      2. pow2 71.3%

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

   8. Applied egg-rr 71.3%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-99} \lor \neg \left(x \leq 6.5 \cdot 10^{-72}\right) \land x \leq 7.2 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in z around 0 69.1%

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

   1. *-commutative 69.1%

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

5. Simplified 69.1%

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

6. Final simplification 69.1%

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

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 89.7%

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

   1. sub-neg 89.7%

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

   2. distribute-lft-neg-in 89.7%

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

   3. *-commutative 89.7%

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

   4. distribute-lft-neg-in 89.7%

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

   5. metadata-eval 89.7%

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

   6. *-commutative 89.7%

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

   7. add-sqr-sqrt 44.9%

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

   8. sqrt-unprod 55.6%

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

   9. swap-sqr 55.6%

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

   10. metadata-eval 55.6%

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

   11. metadata-eval 55.6%

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

   12. swap-sqr 55.6%

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

   13. sqrt-unprod 20.4%

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

   14. add-sqr-sqrt 37.4%

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

   15. expm1-log1p-u 36.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

   16. add-sqr-sqrt 19.9%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

4. Applied egg-rr 65.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

6. Simplified 41.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 42.5%

      \[\leadsto \color{blue}{{x}^{2}} \]

   2. pow2 42.5%

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

8. Applied egg-rr 42.5%

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

9. Final simplification 42.5%

   \[\leadsto x \cdot x \]
10. Add Preprocessing

Developer target: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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