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

Percentage Accurate: 91.1% → 96.7%

Time: 7.0s

Alternatives: 7

Speedup: 0.5×

• Unsound rule application detected in e-graph. Results may not be sound.

• 28.7% 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: 91.1% 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: 96.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+287)))
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (/ t_1 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)) || !(t_1 <= 2e+287)) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else {
		tmp = t_1 / 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)) || !(t_1 <= 2e+287))
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+287]], $MachinePrecision]], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.0000000000000002e287 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 59.3%

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

   2. Taylor expanded in x around 0 56.0%

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

      1. fma-def 56.0%

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

      2. associate-/l* 71.5%

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

      3. associate-/l* 89.9%

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

   4. Simplified 89.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000002e287

   1. Initial program 97.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+287)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ t_1 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)) || !(t_1 <= 2e+287)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / 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) || !(t_1 <= 2e+287)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+287):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / 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)) || !(t_1 <= 2e+287))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / 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) || ~((t_1 <= 2e+287)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / 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[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+287]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.0000000000000002e287 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 59.3%

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

      1. div-sub 56.0%

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

      2. associate-/l* 69.9%

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

      3. associate-/l* 86.7%

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

   3. Applied egg-rr 86.7%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000002e287

   1. Initial program 97.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 3: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(-t\right)}{a}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 4000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z (- t)) a)))
   (if (<= (* x y) -2e+86)
     (* y (/ x a))
     (if (<= (* x y) -1e+33)
       t_1
       (if (<= (* x y) -5e-67)
         (/ (* x y) a)
         (if (<= (* x y) 4000000000.0) t_1 (/ y (/ a x))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * -t) / a;
	double tmp;
	if ((x * y) <= -2e+86) {
		tmp = y * (x / a);
	} else if ((x * y) <= -1e+33) {
		tmp = t_1;
	} else if ((x * y) <= -5e-67) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 4000000000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (z * -t) / a
    if ((x * y) <= (-2d+86)) then
        tmp = y * (x / a)
    else if ((x * y) <= (-1d+33)) then
        tmp = t_1
    else if ((x * y) <= (-5d-67)) then
        tmp = (x * y) / a
    else if ((x * y) <= 4000000000.0d0) then
        tmp = t_1
    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 t_1 = (z * -t) / a;
	double tmp;
	if ((x * y) <= -2e+86) {
		tmp = y * (x / a);
	} else if ((x * y) <= -1e+33) {
		tmp = t_1;
	} else if ((x * y) <= -5e-67) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 4000000000.0) {
		tmp = t_1;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * -t) / a
	tmp = 0
	if (x * y) <= -2e+86:
		tmp = y * (x / a)
	elif (x * y) <= -1e+33:
		tmp = t_1
	elif (x * y) <= -5e-67:
		tmp = (x * y) / a
	elif (x * y) <= 4000000000.0:
		tmp = t_1
	else:
		tmp = y / (a / x)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * Float64(-t)) / a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+86)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= -1e+33)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e-67)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 4000000000.0)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * -t) / a;
	tmp = 0.0;
	if ((x * y) <= -2e+86)
		tmp = y * (x / a);
	elseif ((x * y) <= -1e+33)
		tmp = t_1;
	elseif ((x * y) <= -5e-67)
		tmp = (x * y) / a;
	elseif ((x * y) <= 4000000000.0)
		tmp = t_1;
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+86], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e+33], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e-67], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4000000000.0], t$95$1, N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e86

   1. Initial program 85.5%

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

   2. Taylor expanded in x around inf 78.2%

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

      1. associate-*r/ 88.2%

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

   4. Simplified 88.2%

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

   if -2e86 < (*.f64 x y) < -9.9999999999999995e32 or -4.9999999999999999e-67 < (*.f64 x y) < 4e9

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. associate-*r* 79.6%

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

      3. neg-mul-1 79.6%

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

   4. Simplified 79.6%

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

   if -9.9999999999999995e32 < (*.f64 x y) < -4.9999999999999999e-67

   1. Initial program 95.7%

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

   2. Taylor expanded in x around inf 74.7%

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

   if 4e9 < (*.f64 x y)

   1. Initial program 79.2%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. associate-/l* 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 4000000000:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 4: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* t (/ (- z) a))
   (if (<= (* z t) 2e+295) (/ (- (* x y) (* z t)) a) (* z (/ (- t) a)))))

C

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+295) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = 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) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 2e+295) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = z * (-t / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t * (-z / a)
	elif (z * t) <= 2e+295:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = z * (-t / a)
	return tmp

Julia

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+295)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(z * Float64(Float64(-t) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t * (-z / a);
	elseif ((z * t) <= 2e+295)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = z * (-t / a);
	end
	tmp_2 = tmp;
end

Wolfram

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+295], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.7%

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

   2. Taylor expanded in x around 0 49.7%

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

      1. associate-*r/ 49.7%

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

      2. mul-1-neg 49.7%

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

      3. distribute-rgt-neg-out 49.7%

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

      4. *-commutative 49.7%

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

      5. associate-/l* 88.2%

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

      6. associate-/r/ 88.0%

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

   4. Simplified 88.0%

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

   if -inf.0 < (*.f64 z t) < 2e295

   1. Initial program 94.1%

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

   if 2e295 < (*.f64 z t)

   1. Initial program 56.5%

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

      1. div-inv 56.5%

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

      2. fma-neg 67.9%

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

      3. *-commutative 67.9%

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

      4. distribute-rgt-neg-in 67.9%

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

   3. Applied egg-rr 67.9%

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

   4. Taylor expanded in x around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. associate-*l/ 94.4%

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

      3. distribute-rgt-neg-in 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 5: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;-\frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.4e-97)
   (/ y (/ a x))
   (if (<= y 1.1e-15) (- (/ t (/ a z))) (* y (/ x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.4e-97) {
		tmp = y / (a / x);
	} else if (y <= 1.1e-15) {
		tmp = -(t / (a / z));
	} 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 (y <= (-1.4d-97)) then
        tmp = y / (a / x)
    else if (y <= 1.1d-15) then
        tmp = -(t / (a / z))
    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 (y <= -1.4e-97) {
		tmp = y / (a / x);
	} else if (y <= 1.1e-15) {
		tmp = -(t / (a / z));
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.4e-97:
		tmp = y / (a / x)
	elif y <= 1.1e-15:
		tmp = -(t / (a / z))
	else:
		tmp = y * (x / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.4e-97)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 1.1e-15)
		tmp = Float64(-Float64(t / Float64(a / z)));
	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 (y <= -1.4e-97)
		tmp = y / (a / x);
	elseif (y <= 1.1e-15)
		tmp = -(t / (a / z));
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.4e-97], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-15], (-N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4000000000000001e-97

   1. Initial program 86.9%

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

   2. Taylor expanded in x around inf 49.3%

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

      1. associate-/l* 52.5%

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

   4. Simplified 52.5%

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

   if -1.4000000000000001e-97 < y < 1.09999999999999993e-15

   1. Initial program 92.6%

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

      1. div-inv 92.5%

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

      2. fma-neg 92.5%

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

      3. *-commutative 92.5%

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

      4. distribute-rgt-neg-in 92.5%

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

   3. Applied egg-rr 92.5%

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

   4. Taylor expanded in x around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. associate-/l* 68.9%

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

   6. Simplified 68.9%

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

   if 1.09999999999999993e-15 < y

   1. Initial program 83.6%

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

   2. Taylor expanded in x around inf 70.1%

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

      1. associate-*r/ 75.3%

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

   4. Simplified 75.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;-\frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 6: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-16}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.45e-94)
   (/ y (/ a x))
   (if (<= y 8e-16) (* t (/ (- z) a)) (* y (/ x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.45e-94) {
		tmp = y / (a / x);
	} else if (y <= 8e-16) {
		tmp = t * (-z / 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 (y <= (-1.45d-94)) then
        tmp = y / (a / x)
    else if (y <= 8d-16) then
        tmp = t * (-z / 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 (y <= -1.45e-94) {
		tmp = y / (a / x);
	} else if (y <= 8e-16) {
		tmp = t * (-z / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.45e-94:
		tmp = y / (a / x)
	elif y <= 8e-16:
		tmp = t * (-z / a)
	else:
		tmp = y * (x / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.45e-94)
		tmp = Float64(y / Float64(a / x));
	elseif (y <= 8e-16)
		tmp = Float64(t * Float64(Float64(-z) / 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 (y <= -1.45e-94)
		tmp = y / (a / x);
	elseif (y <= 8e-16)
		tmp = t * (-z / a);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.45e-94], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-16], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999998e-94

   1. Initial program 86.6%

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

   2. Taylor expanded in x around inf 49.2%

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

      1. associate-/l* 52.5%

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

   4. Simplified 52.5%

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

   if -1.44999999999999998e-94 < y < 7.9999999999999998e-16

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. mul-1-neg 70.2%

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

      3. distribute-rgt-neg-out 70.2%

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

      4. *-commutative 70.2%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 69.1%

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

   4. Simplified 69.1%

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

   if 7.9999999999999998e-16 < y

   1. Initial program 83.6%

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

   2. Taylor expanded in x around inf 70.1%

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

      1. associate-*r/ 75.3%

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

   4. Simplified 75.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-16}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 7: 51.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

2. Taylor expanded in x around inf 49.6%

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

   1. associate-*r/ 51.2%

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

4. Simplified 51.2%

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

5. Final simplification 51.2%

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

Developer target: 90.6% 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 2023173 
(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))