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

Percentage Accurate: 90.9% → 95.9%

Time: 7.7s

Alternatives: 11

Speedup: 0.4×

• 28.1% 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: 90.9% 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: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(x \cdot \frac{y}{z \cdot a} - \frac{t}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+270}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y}{z} - t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (* z (- (* x (/ y (* z a))) (/ t a)))
     (if (<= t_1 1e+270) (/ t_1 a) (* (- (* x (/ y z)) t) (/ z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * ((x * (y / (z * a))) - (t / a));
	} else if (t_1 <= 1e+270) {
		tmp = t_1 / a;
	} else {
		tmp = ((x * (y / z)) - t) * (z / a);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z * ((x * (y / (z * a))) - (t / a));
	} else if (t_1 <= 1e+270) {
		tmp = t_1 / a;
	} else {
		tmp = ((x * (y / z)) - t) * (z / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z * ((x * (y / (z * a))) - (t / a))
	elif t_1 <= 1e+270:
		tmp = t_1 / a
	else:
		tmp = ((x * (y / z)) - t) * (z / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(Float64(x * Float64(y / Float64(z * a))) - Float64(t / a)));
	elseif (t_1 <= 1e+270)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(Float64(x * Float64(y / z)) - t) * Float64(z / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z * ((x * (y / (z * a))) - (t / a));
	elseif (t_1 <= 1e+270)
		tmp = t_1 / a;
	else
		tmp = ((x * (y / z)) - t) * (z / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(N[(x * N[(y / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+270], N[(t$95$1 / a), $MachinePrecision], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. associate-/l* 94.8%

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

      5. *-commutative 94.8%

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

   5. Simplified 94.8%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e270

   1. Initial program 98.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1e270 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 73.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 78.5%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in z around inf 87.7%

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{t}{a} + \frac{x \cdot y}{a \cdot z}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. times-frac 92.3%

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

      5. associate-*l/ 92.5%

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

      6. div-sub 97.5%

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

      7. associate-*r/ 90.3%

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

      8. associate-/l* 78.5%

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

      9. *-commutative 78.5%

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

      10. associate-*r/ 81.0%

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

      11. *-rgt-identity 81.0%

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

      12. cancel-sign-sub-inv 81.0%

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

      13. distribute-lft-neg-in 81.0%

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

      14. fma-undefine 81.0%

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

      15. associate-/l* 95.1%

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

   8. Simplified 95.1%

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

Alternative 2: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{-a}\\ t_2 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 10^{+233}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z t) (- a))) (t_2 (/ (- t) (/ a z))))
   (if (<= (* z t) (- INFINITY))
     t_2
     (if (<= (* z t) -2e-53)
       t_1
       (if (<= (* z t) 2e-125)
         (/ (* x y) a)
         (if (<= (* z t) 1e+233) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * t) / -a;
	double t_2 = -t / (a / z);
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_2;
	} else if ((z * t) <= -2e-53) {
		tmp = t_1;
	} else if ((z * t) <= 2e-125) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 1e+233) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * t) / -a;
	double t_2 = -t / (a / z);
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if ((z * t) <= -2e-53) {
		tmp = t_1;
	} else if ((z * t) <= 2e-125) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 1e+233) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * t) / -a
	t_2 = -t / (a / z)
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t_2
	elif (z * t) <= -2e-53:
		tmp = t_1
	elif (z * t) <= 2e-125:
		tmp = (x * y) / a
	elif (z * t) <= 1e+233:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * t) / Float64(-a))
	t_2 = Float64(Float64(-t) / Float64(a / z))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_2;
	elseif (Float64(z * t) <= -2e-53)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-125)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(z * t) <= 1e+233)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * t) / -a;
	t_2 = -t / (a / z);
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t_2;
	elseif ((z * t) <= -2e-53)
		tmp = t_1;
	elseif ((z * t) <= 2e-125)
		tmp = (x * y) / a;
	elseif ((z * t) <= 1e+233)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / (-a)), $MachinePrecision]}, Block[{t$95$2 = N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -2e-53], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-125], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+233], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0 or 9.99999999999999974e232 < (*.f64 z t)

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

   5. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. remove-double-neg 76.5%

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

      4. frac-2neg 76.5%

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

      5. associate-*r/ 94.2%

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

      6. *-commutative 94.2%

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

      7. add-sqr-sqrt 41.4%

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

      8. sqrt-unprod 42.1%

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

      9. sqr-neg 42.1%

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

      10. sqrt-unprod 3.9%

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

      11. add-sqr-sqrt 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 34.7%

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

      3. *-commutative 34.7%

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

      4. clear-num 34.7%

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

      5. div-inv 34.7%

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

      6. *-commutative 34.7%

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

      7. clear-num 34.7%

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

      8. div-inv 34.7%

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

      9. *-un-lft-identity 34.7%

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

      10. associate-*l* 34.7%

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

      11. metadata-eval 34.7%

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

      12. swap-sqr 34.7%

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

      13. clear-num 34.7%

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

      14. div-inv 34.7%

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

      15. clear-num 34.7%

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

      16. div-inv 34.7%

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

      17. sqrt-unprod 41.3%

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

      18. add-sqr-sqrt 94.1%

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

   9. Applied egg-rr 94.3%

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

   if -inf.0 < (*.f64 z t) < -2.00000000000000006e-53 or 2.00000000000000002e-125 < (*.f64 z t) < 9.99999999999999974e232

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.3%

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

      1. mul-1-neg 75.3%

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

      2. *-commutative 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

   5. Simplified 75.3%

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

   if -2.00000000000000006e-53 < (*.f64 z t) < 2.00000000000000002e-125

   1. Initial program 93.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-53}:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 10^{+233}:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{a} - t \cdot \frac{z}{y \cdot a}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+270}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y}{z} - t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (* y (- (/ x a) (* t (/ z (* y a)))))
     (if (<= t_1 1e+270) (/ t_1 a) (* (- (* x (/ y z)) t) (/ z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * ((x / a) - (t * (z / (y * a))));
	} else if (t_1 <= 1e+270) {
		tmp = t_1 / a;
	} else {
		tmp = ((x * (y / z)) - t) * (z / a);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * ((x / a) - (t * (z / (y * a))));
	} else if (t_1 <= 1e+270) {
		tmp = t_1 / a;
	} else {
		tmp = ((x * (y / z)) - t) * (z / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * ((x / a) - (t * (z / (y * a))))
	elif t_1 <= 1e+270:
		tmp = t_1 / a
	else:
		tmp = ((x * (y / z)) - t) * (z / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(Float64(x / a) - Float64(t * Float64(z / Float64(y * a)))));
	elseif (t_1 <= 1e+270)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(Float64(x * Float64(y / z)) - t) * Float64(z / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * ((x / a) - (t * (z / (y * a))));
	elseif (t_1 <= 1e+270)
		tmp = t_1 / a;
	else
		tmp = ((x * (y / z)) - t) * (z / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(x / a), $MachinePrecision] - N[(t * N[(z / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+270], N[(t$95$1 / a), $MachinePrecision], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 69.8%

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

      1. +-commutative 69.8%

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

      2. mul-1-neg 69.8%

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

      3. unsub-neg 69.8%

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

      4. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e270

   1. Initial program 98.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1e270 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 73.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 78.5%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in z around inf 87.7%

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{t}{a} + \frac{x \cdot y}{a \cdot z}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. times-frac 92.3%

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

      5. associate-*l/ 92.5%

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

      6. div-sub 97.5%

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

      7. associate-*r/ 90.3%

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

      8. associate-/l* 78.5%

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

      9. *-commutative 78.5%

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

      10. associate-*r/ 81.0%

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

      11. *-rgt-identity 81.0%

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

      12. cancel-sign-sub-inv 81.0%

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

      13. distribute-lft-neg-in 81.0%

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

      14. fma-undefine 81.0%

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

      15. associate-/l* 95.1%

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

   8. Simplified 95.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{a} - t \cdot \frac{z}{y \cdot a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+270}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y}{z} - t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+184}\right):\\ \;\;\;\;\left(x \cdot \frac{y}{z} - t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 2e+184)))
   (* (- (* x (/ y z)) t) (/ z a))
   (/ (- (* x y) (* z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 2e+184)) {
		tmp = ((x * (y / z)) - t) * (z / a);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 2e+184)) {
		tmp = ((x * (y / z)) - t) * (z / a);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 2e+184):
		tmp = ((x * (y / z)) - t) * (z / a)
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 2e+184))
		tmp = Float64(Float64(Float64(x * Float64(y / z)) - t) * Float64(z / a));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 2e+184)))
		tmp = ((x * (y / z)) - t) * (z / a);
	else
		tmp = ((x * y) - (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+184]], $MachinePrecision]], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 2.00000000000000003e184 < (*.f64 z t)

   1. Initial program 77.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 81.2%

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

      1. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in z around inf 95.0%

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{t}{a} + \frac{x \cdot y}{a \cdot z}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 95.0%

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

      2. mul-1-neg 95.0%

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

      3. unsub-neg 95.0%

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

      4. times-frac 91.6%

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

      5. associate-*l/ 96.6%

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

      6. div-sub 99.8%

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

      7. associate-*r/ 96.7%

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

      8. associate-/l* 81.2%

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

      9. *-commutative 81.2%

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

      10. associate-*r/ 82.9%

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

      11. *-rgt-identity 82.9%

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

      12. cancel-sign-sub-inv 82.9%

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

      13. distribute-lft-neg-in 82.9%

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

      14. fma-undefine 82.9%

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

      15. associate-/l* 99.9%

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 2.00000000000000003e184

   1. Initial program 95.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+184}\right):\\ \;\;\;\;\left(x \cdot \frac{y}{z} - t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+233}\right):\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 1e+233)))
   (/ (- t) (/ a z))
   (/ (- (* x y) (* z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 1e+233)) {
		tmp = -t / (a / z);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 1e+233)) {
		tmp = -t / (a / z);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 1e+233):
		tmp = -t / (a / z)
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 1e+233))
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 1e+233)))
		tmp = -t / (a / z);
	else
		tmp = ((x * y) - (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+233]], $MachinePrecision]], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 9.99999999999999974e232 < (*.f64 z t)

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

   5. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. remove-double-neg 76.5%

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

      4. frac-2neg 76.5%

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

      5. associate-*r/ 94.2%

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

      6. *-commutative 94.2%

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

      7. add-sqr-sqrt 41.4%

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

      8. sqrt-unprod 42.1%

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

      9. sqr-neg 42.1%

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

      10. sqrt-unprod 3.9%

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

      11. add-sqr-sqrt 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 34.7%

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

      3. *-commutative 34.7%

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

      4. clear-num 34.7%

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

      5. div-inv 34.7%

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

      6. *-commutative 34.7%

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

      7. clear-num 34.7%

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

      8. div-inv 34.7%

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

      9. *-un-lft-identity 34.7%

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

      10. associate-*l* 34.7%

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

      11. metadata-eval 34.7%

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

      12. swap-sqr 34.7%

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

      13. clear-num 34.7%

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

      14. div-inv 34.7%

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

      15. clear-num 34.7%

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

      16. div-inv 34.7%

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

      17. sqrt-unprod 41.3%

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

      18. add-sqr-sqrt 94.1%

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

   9. Applied egg-rr 94.3%

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

   if -inf.0 < (*.f64 z t) < 9.99999999999999974e232

   1. Initial program 95.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+233}\right):\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.5 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -3.5e+59)
   (* x (/ y a))
   (if (<= (* x y) 5e+31) (* z (- (/ t a))) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -3.5e+59) {
		tmp = x * (y / a);
	} else if ((x * y) <= 5e+31) {
		tmp = z * -(t / a);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if ((x * y) <= (-3.5d+59)) then
        tmp = x * (y / a)
    else if ((x * y) <= 5d+31) then
        tmp = z * -(t / a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -3.5e+59) {
		tmp = x * (y / a);
	} else if ((x * y) <= 5e+31) {
		tmp = z * -(t / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -3.5e+59:
		tmp = x * (y / a)
	elif (x * y) <= 5e+31:
		tmp = z * -(t / a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -3.5e+59)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= 5e+31)
		tmp = Float64(z * Float64(-Float64(t / a)));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -3.5e+59)
		tmp = x * (y / a);
	elseif ((x * y) <= 5e+31)
		tmp = z * -(t / a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -3.5e+59], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+31], N[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.5e59

   1. Initial program 84.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 64.2%

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

      1. associate-*r/ 71.5%

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

   5. Simplified 71.5%

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

   if -3.5e59 < (*.f64 x y) < 5.00000000000000027e31

   1. Initial program 93.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*r/ 80.5%

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

      3. neg-mul-1 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. distribute-frac-neg 80.5%

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

   5. Simplified 80.5%

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

   if 5.00000000000000027e31 < (*.f64 x y)

   1. Initial program 90.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.5 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+31}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e-16)
   (* y (/ x a))
   (if (<= (* x y) 5e+31) (* t (/ (- z) a)) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e-16) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e+31) {
		tmp = t * (-z / a);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if ((x * y) <= (-5d-16)) then
        tmp = y * (x / a)
    else if ((x * y) <= 5d+31) then
        tmp = t * (-z / a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e-16) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e+31) {
		tmp = t * (-z / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e-16:
		tmp = y * (x / a)
	elif (x * y) <= 5e+31:
		tmp = t * (-z / a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e-16)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 5e+31)
		tmp = Float64(t * Float64(Float64(-z) / a));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e-16)
		tmp = y * (x / a);
	elseif ((x * y) <= 5e+31)
		tmp = t * (-z / a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e-16], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+31], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.0000000000000004e-16

   1. Initial program 87.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 76.7%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in z around 0 61.4%

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

      1. associate-*l/ 66.3%

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

      2. *-commutative 66.3%

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

   8. Simplified 66.3%

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

   if -5.0000000000000004e-16 < (*.f64 x y) < 5.00000000000000027e31

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. associate-/l* 73.7%

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

      3. distribute-rgt-neg-in 73.7%

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

      4. distribute-neg-frac2 73.7%

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

   5. Simplified 73.7%

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

   if 5.00000000000000027e31 < (*.f64 x y)

   1. Initial program 90.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+31}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+117) (* x (/ y a)) (/ (* x y) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+117) {
		tmp = x * (y / a);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if ((x * y) <= (-2d+117)) then
        tmp = x * (y / a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+117) {
		tmp = x * (y / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+117:
		tmp = x * (y / a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+117)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+117)
		tmp = x * (y / a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+117], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000001e117

   1. Initial program 79.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 58.7%

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

      1. associate-*r/ 73.9%

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

   5. Simplified 73.9%

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

   if -2.0000000000000001e117 < (*.f64 x y)

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 42.3%

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

Alternative 9: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.5e-171) (* x (/ y a)) (/ y (/ a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e-171) {
		tmp = x * (y / a);
	} else {
		tmp = y / (a / x);
	}
	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) :: tmp
    if (z <= (-6.5d-171)) then
        tmp = x * (y / a)
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e-171) {
		tmp = x * (y / a);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e-171:
		tmp = x * (y / a)
	else:
		tmp = y / (a / x)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.5e-171)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.5e-171)
		tmp = x * (y / a);
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.5e-171], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000004e-171

   1. Initial program 94.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 41.3%

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

      1. associate-*r/ 39.5%

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

   5. Simplified 39.5%

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

   if -6.5000000000000004e-171 < z

   1. Initial program 88.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 81.3%

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

      1. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in z around 0 47.5%

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

      1. associate-*l/ 46.7%

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

      2. *-commutative 46.7%

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

   8. Simplified 46.7%

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

      1. clear-num 46.2%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{a}{x}}} \]

      2. un-div-inv 46.2%

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

   10. Applied egg-rr 46.2%

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

Alternative 10: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.05e-169) (* x (/ y a)) (* y (/ x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.05e-169) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	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) :: tmp
    if (z <= (-2.05d-169)) then
        tmp = x * (y / a)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.05e-169) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.05e-169:
		tmp = x * (y / a)
	else:
		tmp = y * (x / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.05e-169)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.05e-169)
		tmp = x * (y / a);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.05e-169], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0499999999999999e-169

   1. Initial program 94.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 41.3%

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

      1. associate-*r/ 39.5%

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

   5. Simplified 39.5%

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

   if -2.0499999999999999e-169 < z

   1. Initial program 88.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 81.3%

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

      1. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in z around 0 47.5%

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

      1. associate-*l/ 46.7%

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

      2. *-commutative 46.7%

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

   8. Simplified 46.7%

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

Alternative 11: 51.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x * (y / 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 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 44.7%

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

   1. associate-*r/ 44.3%

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

5. Simplified 44.3%

   \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]
6. Add Preprocessing

Developer Target 1: 91.3% accurate, 0.4× 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 2024123 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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