Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 92.0% → 95.2%

Time: 7.1s

Alternatives: 7

Speedup: 0.5×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \left(-z\right) \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* t (/ (- z) a))
   (if (<= (* z t) 2e+217)
     (/ (fma x y (* z (- t))) a)
     (fma x (/ y a) (* (- z) (/ t a))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 2e+217) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = fma(x, (y / a), (-z * (t / a)));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(-z) / a));
	elseif (Float64(z * t) <= 2e+217)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(x, Float64(y / a), Float64(Float64(-z) * Float64(t / a)));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+217], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision] + N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 68.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 72.8%

         \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      2. associate-*l/ 95.2%

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

      3. associate-*r* 95.2%

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

      4. neg-mul-1 95.2%

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

      5. distribute-frac-neg 95.2%

         \[\leadsto \color{blue}{\frac{-z}{a}} \cdot t \]

   4. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]

   if -inf.0 < (*.f64 z t) < 1.99999999999999992e217

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. fma-neg 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

   3. Simplified 98.8%

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

   if 1.99999999999999992e217 < (*.f64 z t)

   1. Initial program 67.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 64.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative 64.3%

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

      2. mul-1-neg 64.3%

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

      3. unsub-neg 64.3%

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

      4. associate-*l/ 61.2%

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

      5. *-commutative 61.2%

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

      6. div-inv 61.2%

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

      7. associate-*l* 93.5%

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

      8. div-inv 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. associate-*r/ 61.2%

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

      2. add-sqr-sqrt 41.2%

         \[\leadsto y \cdot \frac{x}{a} - \frac{t \cdot z}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

      3. sqrt-unprod 31.8%

         \[\leadsto y \cdot \frac{x}{a} - \frac{t \cdot z}{\color{blue}{\sqrt{a \cdot a}}} \]

      4. sqr-neg 31.8%

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

      5. sqrt-unprod 0.1%

         \[\leadsto y \cdot \frac{x}{a} - \frac{t \cdot z}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

      6. add-sqr-sqrt 3.4%

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

      7. *-commutative 3.4%

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

      8. associate-/l* 3.5%

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

      9. add-sqr-sqrt 0.2%

         \[\leadsto y \cdot \frac{x}{a} - \frac{z}{\frac{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{t}} \]

      10. sqrt-unprod 38.4%

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

      11. sqr-neg 38.4%

          \[\leadsto y \cdot \frac{x}{a} - \frac{z}{\frac{\sqrt{\color{blue}{a \cdot a}}}{t}} \]

      12. sqrt-unprod 52.8%

          \[\leadsto y \cdot \frac{x}{a} - \frac{z}{\frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{t}} \]

      13. add-sqr-sqrt 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in y around 0 64.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 64.3%

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

      2. +-commutative 64.3%

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

      3. mul-1-neg 64.3%

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

      4. associate-*l/ 96.6%

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

      5. fma-udef 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \left(-z\right) \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 2: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* t (/ (- z) a))
   (if (<= (* z t) 2e+259) (/ (fma x y (* z (- t))) a) (/ (- t) (/ a z)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 2e+259) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(-z) / a));
	elseif (Float64(z * t) <= 2e+259)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+259], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 68.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 72.8%

         \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      2. associate-*l/ 95.2%

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

      3. associate-*r* 95.2%

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

      4. neg-mul-1 95.2%

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

      5. distribute-frac-neg 95.2%

         \[\leadsto \color{blue}{\frac{-z}{a}} \cdot t \]

   4. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]

   if -inf.0 < (*.f64 z t) < 2e259

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. fma-neg 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

   3. Simplified 98.8%

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

   if 2e259 < (*.f64 z t)

   1. Initial program 60.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 60.0%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. associate-/l* 99.9%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 3: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* t (/ (- z) a))
   (if (<= (* z t) 2e+259) (/ (- (* x y) (* z t)) a) (/ (- t) (/ a z)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 2e+259) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 2e+259) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t * (-z / a)
	elif (z * t) <= 2e+259:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -t / (a / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(-z) / a));
	elseif (Float64(z * t) <= 2e+259)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t * (-z / a);
	elseif ((z * t) <= 2e+259)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -t / (a / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+259], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 68.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 72.8%

         \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      2. associate-*l/ 95.2%

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

      3. associate-*r* 95.2%

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

      4. neg-mul-1 95.2%

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

      5. distribute-frac-neg 95.2%

         \[\leadsto \color{blue}{\frac{-z}{a}} \cdot t \]

   4. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]

   if -inf.0 < (*.f64 z t) < 2e259

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 2e259 < (*.f64 z t)

   1. Initial program 60.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 60.0%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. associate-/l* 99.9%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 4: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+16}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+60)
   (/ (* x y) a)
   (if (<= (* x y) 1e+16) (/ (- t) (/ a z)) (* x (/ y a)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+60) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 1e+16) {
		tmp = -t / (a / z);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+60)) then
        tmp = (x * y) / a
    else if ((x * y) <= 1d+16) then
        tmp = -t / (a / z)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+60) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 1e+16) {
		tmp = -t / (a / z);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+60:
		tmp = (x * y) / a
	elif (x * y) <= 1e+16:
		tmp = -t / (a / z)
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+60)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 1e+16)
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+60)
		tmp = (x * y) / a;
	elseif ((x * y) <= 1e+16)
		tmp = -t / (a / z);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+60], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+16], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.9999999999999999e60

   1. Initial program 92.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 78.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if -1.9999999999999999e60 < (*.f64 x y) < 1e16

   1. Initial program 92.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 75.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 75.7%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. associate-/l* 76.5%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

   4. Simplified 76.5%

      \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]

   if 1e16 < (*.f64 x y)

   1. Initial program 91.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 89.4%

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

   3. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

   5. Simplified 74.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+16}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5: 51.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y 1.3e-153) (* y (/ x a)) (* x (/ y a))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.3e-153) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= 1.3d-153) then
        tmp = y * (x / a)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.3e-153) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= 1.3e-153:
		tmp = y * (x / a)
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 1.3e-153)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 1.3e-153)
		tmp = y * (x / a);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, 1.3e-153], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-153

   1. Initial program 91.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 44.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 41.7%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   if 1.3000000000000001e-153 < y

   1. Initial program 94.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 91.8%

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

   3. Taylor expanded in t around 0 51.8%

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

      1. associate-*r/ 52.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

   5. Simplified 52.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6: 51.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ x \cdot \frac{y}{a} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* x (/ y a)))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return x * (y / a);
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x * (y / a)
end function

Java

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return x * (y / a)

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(x * Float64(y / a))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = x * (y / a);
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around 0 90.4%

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

3. Taylor expanded in t around 0 46.5%

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

   1. associate-*r/ 45.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

5. Simplified 45.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

6. Final simplification 45.5%

   \[\leadsto x \cdot \frac{y}{a} \]

Alternative 7: 51.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{x \cdot y}{a} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (/ (* x y) a))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return (x * y) / a;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x * y) / a
end function

Java

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return (x * y) / a

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(Float64(x * y) / a)
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = (x * y) / a;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 46.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

3. Final simplification 46.5%

   \[\leadsto \frac{x \cdot y}{a} \]

Developer target: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023318 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))