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

Percentage Accurate: 90.5% → 96.1%

Time: 10.2s

Alternatives: 7

Speedup: 0.7×

• 48.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+300)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* (/ y (/ 1.0 z)) (/ 4.0 (/ 1.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+300) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((y / (1.0 / z)) * (4.0 / (1.0 / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4d+300) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - ((y / (1.0d0 / z)) * (4.0d0 / (1.0d0 / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+300) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((y / (1.0 / z)) * (4.0 / (1.0 / z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e+300:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - ((y / (1.0 / z)) * (4.0 / (1.0 / z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+300)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(y / Float64(1.0 / z)) * Float64(4.0 / Float64(1.0 / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e+300)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - ((y / (1.0 / z)) * (4.0 / (1.0 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+300], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(y / N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000002e300

   1. Initial program 97.9%

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

   if 4.0000000000000002e300 < (*.f64 z z)

   1. Initial program 77.6%

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

      1. sub-neg 77.6%

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

      2. flip-+ 0.0%

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

      3. pow2 0.0%

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

      4. pow2 0.0%

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

      5. pow-prod-up 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow2 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. un-div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow-sqr 0.0%

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

      6. sqr-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. remove-double-neg 0.0%

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

      9. flip-- 77.6%

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

      10. unpow2 77.6%

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

      11. fma-neg 77.6%

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

      12. add-sqr-sqrt 29.7%

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

      13. sqrt-prod 76.1%

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

      14. sqr-neg 76.1%

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

      15. sqrt-prod 47.9%

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

      16. add-sqr-sqrt 77.6%

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

   5. Applied egg-rr 77.6%

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

   6. Taylor expanded in z around inf 77.6%

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

      1. add-sqr-sqrt 77.6%

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

      2. times-frac 77.6%

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

      3. sqrt-div 77.6%

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

      4. metadata-eval 77.6%

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

      5. unpow2 77.6%

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

      6. sqrt-prod 38.1%

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

      7. add-sqr-sqrt 41.4%

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

      8. sqrt-div 41.4%

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

      9. metadata-eval 41.4%

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

      10. unpow2 41.4%

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

      11. sqrt-prod 47.5%

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

      12. add-sqr-sqrt 90.0%

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

   8. Applied egg-rr 90.0%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

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

Alternative 2: 92.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. fma-neg 95.4%

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

   2. distribute-lft-neg-in 95.4%

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

   3. *-commutative 95.4%

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

   4. distribute-rgt-neg-in 95.4%

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

   5. metadata-eval 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 3: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \frac{y \cdot 4}{\frac{1}{t}}\\ \end{array} \end{array} \]

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) (/ (* y 4.0) (/ 1.0 t))))))

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) - ((y * 4.0) / (1.0 / t));
	}
	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) - ((y * 4.0) / (1.0 / t));
	}
	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) - ((y * 4.0) / (1.0 / t))
	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(Float64(x * x) - Float64(Float64(y * 4.0) / Float64(1.0 / t)));
	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) - ((y * 4.0) / (1.0 / t));
	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[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.6%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.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. Step-by-step derivation

      1. sub-neg 0.0%

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

      2. flip-+ 0.0%

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

      3. pow2 0.0%

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

      4. pow2 0.0%

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

      5. pow-prod-up 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow2 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. un-div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow-sqr 0.0%

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

      6. sqr-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. remove-double-neg 0.0%

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

      9. flip-- 0.0%

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

      10. unpow2 0.0%

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

      11. fma-neg 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-prod 8.3%

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

      14. sqr-neg 8.3%

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

      15. sqrt-prod 8.3%

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

      16. add-sqr-sqrt 8.3%

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

   5. Applied egg-rr 8.3%

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

   6. Taylor expanded in z around 0 50.0%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) \leq \infty:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \frac{y \cdot 4}{\frac{1}{t}}\\ \end{array} \]

Alternative 4: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \frac{y \cdot 4}{\frac{1}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.45e+102)
   (- (* x x) (* -4.0 (* t y)))
   (- (* x x) (/ (* y 4.0) (/ 1.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+102) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - ((y * 4.0) / (1.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.45d+102) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = (x * x) - ((y * 4.0d0) / (1.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+102) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - ((y * 4.0) / (1.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.45e+102:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = (x * x) - ((y * 4.0) / (1.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.45e+102)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) / Float64(1.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.45e+102)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = (x * x) - ((y * 4.0) / (1.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.45e+102], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4500000000000001e102

   1. Initial program 95.2%

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

   2. Taylor expanded in z around 0 67.5%

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

      1. *-commutative 67.5%

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

   4. Simplified 67.5%

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

   if 1.4500000000000001e102 < z

   1. Initial program 82.7%

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

      1. sub-neg 82.7%

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

      2. flip-+ 7.3%

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

      3. pow2 7.3%

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

      4. pow2 7.3%

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

      5. pow-prod-up 7.3%

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

      6. metadata-eval 7.3%

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

      7. pow2 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. clear-num 7.3%

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

      2. un-div-inv 7.3%

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

      3. clear-num 7.3%

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

      4. metadata-eval 7.3%

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

      5. pow-sqr 7.3%

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

      6. sqr-neg 7.3%

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

      7. sub-neg 7.3%

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

      8. remove-double-neg 7.3%

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

      9. flip-- 82.8%

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

      10. unpow2 82.8%

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

      11. fma-neg 82.8%

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

      12. add-sqr-sqrt 42.4%

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

      13. sqrt-prod 72.5%

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

      14. sqr-neg 72.5%

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

      15. sqrt-prod 40.3%

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

      16. add-sqr-sqrt 80.5%

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

   5. Applied egg-rr 80.5%

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

   6. Taylor expanded in z around 0 15.5%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \frac{y \cdot 4}{\frac{1}{t}}\\ \end{array} \]

Alternative 5: 66.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

2. Taylor expanded in z around 0 58.5%

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

   1. *-commutative 58.5%

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

4. Simplified 58.5%

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

5. Final simplification 58.5%

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

Alternative 6: 31.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1.89999999999999989e102

   1. Initial program 95.2%

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

   2. Taylor expanded in t around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-*l* 32.7%

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

   4. Simplified 32.7%

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

   if 1.89999999999999989e102 < z

   1. Initial program 82.7%

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

   2. Taylor expanded in t around inf 7.6%

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

      1. *-commutative 7.6%

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

      2. *-commutative 7.6%

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

   4. Simplified 7.6%

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

      1. add-sqr-sqrt 3.6%

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

      2. sqrt-unprod 12.8%

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

      3. *-commutative 12.8%

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

      4. *-commutative 12.8%

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

      5. swap-sqr 12.8%

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

      6. metadata-eval 12.8%

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

      7. metadata-eval 12.8%

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

      8. swap-sqr 12.8%

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

      9. *-commutative 12.8%

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

      10. *-commutative 12.8%

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

      11. sqrt-unprod 3.1%

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

      12. add-log-exp 11.6%

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

      13. add-sqr-sqrt 19.2%

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

      14. associate-*r* 19.2%

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

      15. exp-prod 12.6%

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

   6. Applied egg-rr 12.6%

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

      1. log-pow 12.6%

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

      2. rem-log-exp 6.5%

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

   8. Simplified 6.5%

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

4. Final simplification 28.1%

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

Alternative 7: 6.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

2. Taylor expanded in t around inf 28.3%

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

   1. *-commutative 28.3%

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

   2. *-commutative 28.3%

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

4. Simplified 28.3%

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

   1. add-sqr-sqrt 15.4%

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

   2. sqrt-unprod 20.0%

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

   3. *-commutative 20.0%

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

   4. *-commutative 20.0%

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

   5. swap-sqr 20.0%

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

   6. metadata-eval 20.0%

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

   7. metadata-eval 20.0%

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

   8. swap-sqr 20.0%

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

   9. *-commutative 20.0%

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

   10. *-commutative 20.0%

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

   11. sqrt-unprod 4.7%

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

   12. add-log-exp 11.4%

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

   13. add-sqr-sqrt 17.1%

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

   14. associate-*r* 17.1%

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

   15. exp-prod 14.7%

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

6. Applied egg-rr 14.7%

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

   1. log-pow 14.4%

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

   2. rem-log-exp 7.9%

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

8. Simplified 7.9%

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

9. Final simplification 7.9%

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

Developer target: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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