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

Percentage Accurate: 91.6% → 96.3%

Time: 13.4s

Alternatives: 11

Speedup: 0.3×

• 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: 91.6% 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: 96.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := t \cdot \frac{z}{a\_m}\\ t_2 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{x}{\sqrt{a\_m}}}{\sqrt{a\_m}} - t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{\sqrt{a\_m}}\right) \cdot \frac{y}{\sqrt{a\_m}} - t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* t (/ z a_m))) (t_2 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_2 (- INFINITY))
      (- (/ (* y (/ x (sqrt a_m))) (sqrt a_m)) t_1)
      (if (<= t_2 5e+261)
        (/ (fma x y (* z (- t))) a_m)
        (- (* (* x (/ 1.0 (sqrt a_m))) (/ y (sqrt a_m))) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z / a_m);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((y * (x / sqrt(a_m))) / sqrt(a_m)) - t_1;
	} else if (t_2 <= 5e+261) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = ((x * (1.0 / sqrt(a_m))) * (y / sqrt(a_m))) - t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(t * Float64(z / a_m))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * Float64(x / sqrt(a_m))) / sqrt(a_m)) - t_1);
	elseif (t_2 <= 5e+261)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 / sqrt(a_m))) * Float64(y / sqrt(a_m))) - t_1);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y * N[(x / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+261], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(N[(x * N[(1.0 / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

      1. div-sub 60.0%

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

      2. *-un-lft-identity 60.0%

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

      3. add-sqr-sqrt 23.5%

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

      4. times-frac 23.5%

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

      5. fma-neg 23.5%

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

      6. associate-/l* 35.1%

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

   4. Applied egg-rr 35.1%

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

      1. fma-undefine 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. cancel-sign-sub-inv 35.1%

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

      4. associate-*l/ 35.1%

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

      5. *-lft-identity 35.1%

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

      6. *-commutative 35.1%

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

      7. associate-/l* 43.4%

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

      8. associate-*r/ 31.8%

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

      9. *-commutative 31.8%

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

      10. associate-/l* 43.3%

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

   6. Simplified 43.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000001e261

   1. Initial program 98.7%

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

      1. div-sub 97.1%

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

      2. *-commutative 97.1%

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

      3. div-sub 98.7%

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

      4. *-commutative 98.7%

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

      5. fma-neg 98.7%

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

      6. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   if 5.0000000000000001e261 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 71.5%

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

      1. div-sub 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. add-sqr-sqrt 32.9%

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

      4. times-frac 32.9%

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

      5. fma-neg 32.9%

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

      6. associate-/l* 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. fma-undefine 37.5%

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

      2. distribute-lft-neg-in 37.5%

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

      3. cancel-sign-sub-inv 37.5%

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

      4. associate-/l* 39.9%

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

      5. associate-*r* 39.9%

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

      6. associate-*r/ 35.3%

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

      7. *-commutative 35.3%

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

      8. associate-/l* 40.0%

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

   6. Simplified 40.0%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{a}} \cdot x\right) \cdot \frac{y}{\sqrt{a}} - t \cdot \frac{z}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{x}{\sqrt{a}}}{\sqrt{a}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{\sqrt{a}}\right) \cdot \frac{y}{\sqrt{a}} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{x}{\sqrt{a\_m}}}{\sqrt{a\_m}} - t \cdot \frac{z}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m} - \frac{t}{\frac{a\_m}{z}}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (- (/ (* y (/ x (sqrt a_m))) (sqrt a_m)) (* t (/ z a_m)))
      (if (<= t_1 1e+168)
        (/ (fma x y (* z (- t))) a_m)
        (- (* x (/ y a_m)) (/ t (/ a_m z))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((y * (x / sqrt(a_m))) / sqrt(a_m)) - (t * (z / a_m));
	} else if (t_1 <= 1e+168) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = (x * (y / a_m)) - (t / (a_m / z));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * Float64(x / sqrt(a_m))) / sqrt(a_m)) - Float64(t * Float64(z / a_m)));
	elseif (t_1 <= 1e+168)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(t / Float64(a_m / z)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * N[(x / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+168], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

      1. div-sub 60.0%

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

      2. *-un-lft-identity 60.0%

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

      3. add-sqr-sqrt 23.5%

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

      4. times-frac 23.5%

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

      5. fma-neg 23.5%

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

      6. associate-/l* 35.1%

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

   4. Applied egg-rr 35.1%

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

      1. fma-undefine 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. cancel-sign-sub-inv 35.1%

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

      4. associate-*l/ 35.1%

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

      5. *-lft-identity 35.1%

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

      6. *-commutative 35.1%

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

      7. associate-/l* 43.4%

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

      8. associate-*r/ 31.8%

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

      9. *-commutative 31.8%

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

      10. associate-/l* 43.3%

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

   6. Simplified 43.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999993e167

   1. Initial program 98.6%

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

      1. div-sub 96.8%

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

      2. *-commutative 96.8%

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

      3. div-sub 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-neg 98.6%

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

      6. distribute-rgt-neg-out 98.6%

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

   3. Simplified 98.6%

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

   if 9.9999999999999993e167 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      2. *-un-lft-identity 81.2%

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

      3. add-sqr-sqrt 37.9%

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

      4. times-frac 37.8%

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

      5. fma-neg 37.8%

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

      6. associate-/l* 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. fma-undefine 40.8%

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

      2. distribute-lft-neg-in 40.8%

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

      3. cancel-sign-sub-inv 40.8%

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

      4. associate-*l/ 40.8%

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

      5. *-lft-identity 40.8%

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

      6. *-commutative 40.8%

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

      7. associate-/l* 42.4%

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

      8. associate-*r/ 39.4%

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

      9. *-commutative 39.4%

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

      10. associate-/l* 42.4%

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

   6. Simplified 42.4%

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

      1. clear-num 42.4%

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

      2. un-div-inv 42.4%

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

   8. Applied egg-rr 42.4%

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

      1. associate-/l* 42.5%

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

      2. associate-/l/ 42.5%

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

      3. add-sqr-sqrt 96.6%

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

      4. associate-/l* 88.9%

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

      5. associate-*l/ 98.1%

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

   10. Applied egg-rr 98.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{x}{\sqrt{a}}}{\sqrt{a}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ t_2 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1 - z \cdot \frac{t}{a\_m}\\ \mathbf{elif}\;t\_2 \leq 10^{+168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{t}{\frac{a\_m}{z}}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* x (/ y a_m))) (t_2 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_2 (- INFINITY))
      (- t_1 (* z (/ t a_m)))
      (if (<= t_2 1e+168)
        (/ (fma x y (* z (- t))) a_m)
        (- t_1 (/ t (/ a_m z))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * (y / a_m);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1 - (z * (t / a_m));
	} else if (t_2 <= 1e+168) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = t_1 - (t / (a_m / z));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(x * Float64(y / a_m))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_1 - Float64(z * Float64(t / a_m)));
	elseif (t_2 <= 1e+168)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = Float64(t_1 - Float64(t / Float64(a_m / z)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(t$95$1 - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+168], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(t$95$1 - N[(t / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

      1. div-sub 60.0%

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

      2. associate-/l* 75.9%

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

      3. associate-/l* 95.5%

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

   4. Applied egg-rr 95.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999993e167

   1. Initial program 98.6%

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

      1. div-sub 96.8%

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

      2. *-commutative 96.8%

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

      3. div-sub 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-neg 98.6%

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

      6. distribute-rgt-neg-out 98.6%

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

   3. Simplified 98.6%

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

   if 9.9999999999999993e167 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      2. *-un-lft-identity 81.2%

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

      3. add-sqr-sqrt 37.9%

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

      4. times-frac 37.8%

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

      5. fma-neg 37.8%

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

      6. associate-/l* 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. fma-undefine 40.8%

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

      2. distribute-lft-neg-in 40.8%

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

      3. cancel-sign-sub-inv 40.8%

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

      4. associate-*l/ 40.8%

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

      5. *-lft-identity 40.8%

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

      6. *-commutative 40.8%

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

      7. associate-/l* 42.4%

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

      8. associate-*r/ 39.4%

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

      9. *-commutative 39.4%

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

      10. associate-/l* 42.4%

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

   6. Simplified 42.4%

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

      1. clear-num 42.4%

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

      2. un-div-inv 42.4%

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

   8. Applied egg-rr 42.4%

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

      1. associate-/l* 42.5%

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

      2. associate-/l/ 42.5%

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

      3. add-sqr-sqrt 96.6%

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

      4. associate-/l* 88.9%

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

      5. associate-*l/ 98.1%

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

   10. Applied egg-rr 98.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+303}\right):\\ \;\;\;\;x \cdot \frac{y}{a\_m} - z \cdot \frac{t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+303)))
      (- (* x (/ y a_m)) (* z (/ t a_m)))
      (/ t_1 a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+303)) {
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	} else {
		tmp = t_1 / a_m;
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+303)) {
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	} else {
		tmp = t_1 / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+303):
		tmp = (x * (y / a_m)) - (z * (t / a_m))
	else:
		tmp = t_1 / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+303))
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(z * Float64(t / a_m)));
	else
		tmp = Float64(t_1 / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+303)))
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	else
		tmp = t_1 / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+303]], $MachinePrecision]], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1e303 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.3%

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

      1. div-sub 63.3%

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

      2. associate-/l* 76.6%

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

      3. associate-/l* 96.1%

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

   4. Applied egg-rr 96.1%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e303

   1. Initial program 98.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+303}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ t_2 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1 - z \cdot \frac{t}{a\_m}\\ \mathbf{elif}\;t\_2 \leq 10^{+168}:\\ \;\;\;\;\frac{t\_2}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{t}{\frac{a\_m}{z}}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* x (/ y a_m))) (t_2 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_2 (- INFINITY))
      (- t_1 (* z (/ t a_m)))
      (if (<= t_2 1e+168) (/ t_2 a_m) (- t_1 (/ t (/ a_m z))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * (y / a_m);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1 - (z * (t / a_m));
	} else if (t_2 <= 1e+168) {
		tmp = t_2 / a_m;
	} else {
		tmp = t_1 - (t / (a_m / z));
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * (y / a_m);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 - (z * (t / a_m));
	} else if (t_2 <= 1e+168) {
		tmp = t_2 / a_m;
	} else {
		tmp = t_1 - (t / (a_m / z));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = x * (y / a_m)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1 - (z * (t / a_m))
	elif t_2 <= 1e+168:
		tmp = t_2 / a_m
	else:
		tmp = t_1 - (t / (a_m / z))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(x * Float64(y / a_m))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_1 - Float64(z * Float64(t / a_m)));
	elseif (t_2 <= 1e+168)
		tmp = Float64(t_2 / a_m);
	else
		tmp = Float64(t_1 - Float64(t / Float64(a_m / z)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = x * (y / a_m);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1 - (z * (t / a_m));
	elseif (t_2 <= 1e+168)
		tmp = t_2 / a_m;
	else
		tmp = t_1 - (t / (a_m / z));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(t$95$1 - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+168], N[(t$95$2 / a$95$m), $MachinePrecision], N[(t$95$1 - N[(t / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

      1. div-sub 60.0%

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

      2. associate-/l* 75.9%

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

      3. associate-/l* 95.5%

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

   4. Applied egg-rr 95.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999993e167

   1. Initial program 98.6%

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

   if 9.9999999999999993e167 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      2. *-un-lft-identity 81.2%

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

      3. add-sqr-sqrt 37.9%

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

      4. times-frac 37.8%

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

      5. fma-neg 37.8%

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

      6. associate-/l* 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. fma-undefine 40.8%

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

      2. distribute-lft-neg-in 40.8%

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

      3. cancel-sign-sub-inv 40.8%

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

      4. associate-*l/ 40.8%

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

      5. *-lft-identity 40.8%

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

      6. *-commutative 40.8%

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

      7. associate-/l* 42.4%

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

      8. associate-*r/ 39.4%

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

      9. *-commutative 39.4%

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

      10. associate-/l* 42.4%

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

   6. Simplified 42.4%

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

      1. clear-num 42.4%

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

      2. un-div-inv 42.4%

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

   8. Applied egg-rr 42.4%

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

      1. associate-/l* 42.5%

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

      2. associate-/l/ 42.5%

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

      3. add-sqr-sqrt 96.6%

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

      4. associate-/l* 88.9%

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

      5. associate-*l/ 98.1%

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

   10. Applied egg-rr 98.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+168}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-85} \lor \neg \left(t \leq 2 \cdot 10^{+170}\right):\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= t -2.7e-85) (not (<= t 2e+170)))
    (* t (/ z (- a_m)))
    (* y (/ x a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((t <= -2.7e-85) || !(t <= 2e+170)) {
		tmp = t * (z / -a_m);
	} else {
		tmp = y * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((t <= (-2.7d-85)) .or. (.not. (t <= 2d+170))) then
        tmp = t * (z / -a_m)
    else
        tmp = y * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((t <= -2.7e-85) || !(t <= 2e+170)) {
		tmp = t * (z / -a_m);
	} else {
		tmp = y * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (t <= -2.7e-85) or not (t <= 2e+170):
		tmp = t * (z / -a_m)
	else:
		tmp = y * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((t <= -2.7e-85) || !(t <= 2e+170))
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	else
		tmp = Float64(y * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((t <= -2.7e-85) || ~((t <= 2e+170)))
		tmp = t * (z / -a_m);
	else
		tmp = y * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[t, -2.7e-85], N[Not[LessEqual[t, 2e+170]], $MachinePrecision]], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.7000000000000001e-85 or 2.00000000000000007e170 < t

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. associate-/l* 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

   5. Simplified 68.3%

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

   if -2.7000000000000001e-85 < t < 2.00000000000000007e170

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r/ 69.1%

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

   5. Simplified 69.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-85} \lor \neg \left(t \leq 2 \cdot 10^{+170}\right):\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-85}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= t -2.45e-85)
    (* z (/ t (- a_m)))
    (if (<= t 7.2e+171) (* y (/ x a_m)) (* t (/ z (- a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= -2.45e-85) {
		tmp = z * (t / -a_m);
	} else if (t <= 7.2e+171) {
		tmp = y * (x / a_m);
	} else {
		tmp = t * (z / -a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (t <= (-2.45d-85)) then
        tmp = z * (t / -a_m)
    else if (t <= 7.2d+171) then
        tmp = y * (x / a_m)
    else
        tmp = t * (z / -a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= -2.45e-85) {
		tmp = z * (t / -a_m);
	} else if (t <= 7.2e+171) {
		tmp = y * (x / a_m);
	} else {
		tmp = t * (z / -a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if t <= -2.45e-85:
		tmp = z * (t / -a_m)
	elif t <= 7.2e+171:
		tmp = y * (x / a_m)
	else:
		tmp = t * (z / -a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (t <= -2.45e-85)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	elseif (t <= 7.2e+171)
		tmp = Float64(y * Float64(x / a_m));
	else
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (t <= -2.45e-85)
		tmp = z * (t / -a_m);
	elseif (t <= 7.2e+171)
		tmp = y * (x / a_m);
	else
		tmp = t * (z / -a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[t, -2.45e-85], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e+171], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -2.45000000000000007e-85

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*r/ 66.0%

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

      3. neg-mul-1 66.0%

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

      4. distribute-lft-neg-in 66.0%

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

   5. Simplified 66.0%

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

   if -2.45000000000000007e-85 < t < 7.20000000000000036e171

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r/ 69.1%

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

   5. Simplified 69.1%

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

   if 7.20000000000000036e171 < t

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. associate-/l* 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

   5. Simplified 81.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-85}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a\_m}{-z}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= t -2.2e-80)
    (* z (/ t (- a_m)))
    (if (<= t 2e+170) (* y (/ x a_m)) (/ t (/ a_m (- z)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= -2.2e-80) {
		tmp = z * (t / -a_m);
	} else if (t <= 2e+170) {
		tmp = y * (x / a_m);
	} else {
		tmp = t / (a_m / -z);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (t <= (-2.2d-80)) then
        tmp = z * (t / -a_m)
    else if (t <= 2d+170) then
        tmp = y * (x / a_m)
    else
        tmp = t / (a_m / -z)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= -2.2e-80) {
		tmp = z * (t / -a_m);
	} else if (t <= 2e+170) {
		tmp = y * (x / a_m);
	} else {
		tmp = t / (a_m / -z);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if t <= -2.2e-80:
		tmp = z * (t / -a_m)
	elif t <= 2e+170:
		tmp = y * (x / a_m)
	else:
		tmp = t / (a_m / -z)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (t <= -2.2e-80)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	elseif (t <= 2e+170)
		tmp = Float64(y * Float64(x / a_m));
	else
		tmp = Float64(t / Float64(a_m / Float64(-z)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (t <= -2.2e-80)
		tmp = z * (t / -a_m);
	elseif (t <= 2e+170)
		tmp = y * (x / a_m);
	else
		tmp = t / (a_m / -z);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[t, -2.2e-80], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+170], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(t / N[(a$95$m / (-z)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -2.2000000000000001e-80

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*r/ 66.0%

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

      3. neg-mul-1 66.0%

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

      4. distribute-lft-neg-in 66.0%

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

   5. Simplified 66.0%

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

   if -2.2000000000000001e-80 < t < 2.00000000000000007e170

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r/ 69.1%

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

   5. Simplified 69.1%

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

   if 2.00000000000000007e170 < t

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. *-commutative 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   5. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-/l* 81.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 7.5%

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

      5. sqr-neg 7.5%

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

      6. sqrt-unprod 1.1%

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

      7. add-sqr-sqrt 1.1%

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

      8. clear-num 1.1%

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

      9. div-inv 1.1%

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

      10. frac-2neg 1.1%

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

      11. div-inv 1.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 32.9%

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

      14. sqr-neg 32.9%

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

      15. sqrt-unprod 81.5%

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

      16. add-sqr-sqrt 81.7%

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

      17. distribute-neg-frac2 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+242}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= z -1.25e+242) (* z (/ t (- a_m))) (/ (- (* x y) (* z t)) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (z <= -1.25e+242) {
		tmp = z * (t / -a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (z <= (-1.25d+242)) then
        tmp = z * (t / -a_m)
    else
        tmp = ((x * y) - (z * t)) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (z <= -1.25e+242) {
		tmp = z * (t / -a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if z <= -1.25e+242:
		tmp = z * (t / -a_m)
	else:
		tmp = ((x * y) - (z * t)) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (z <= -1.25e+242)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (z <= -1.25e+242)
		tmp = z * (t / -a_m);
	else
		tmp = ((x * y) - (z * t)) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[z, -1.25e+242], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.2500000000000001e242

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*r/ 96.4%

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

      3. neg-mul-1 96.4%

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

      4. distribute-lft-neg-in 96.4%

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

   5. Simplified 96.4%

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

   if -1.2500000000000001e242 < z

   1. Initial program 92.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+242}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-208}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= x -4.7e-208) (* y (/ x a_m)) (* x (/ y a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (x <= -4.7e-208) {
		tmp = y * (x / a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (x <= (-4.7d-208)) then
        tmp = y * (x / a_m)
    else
        tmp = x * (y / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (x <= -4.7e-208) {
		tmp = y * (x / a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if x <= -4.7e-208:
		tmp = y * (x / a_m)
	else:
		tmp = x * (y / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (x <= -4.7e-208)
		tmp = Float64(y * Float64(x / a_m));
	else
		tmp = Float64(x * Float64(y / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (x <= -4.7e-208)
		tmp = y * (x / a_m);
	else
		tmp = x * (y / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[x, -4.7e-208], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000003e-208

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-*r/ 53.6%

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

   5. Simplified 53.6%

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

   if -4.7000000000000003e-208 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf 50.0%

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

      1. associate-*r/ 55.4%

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

   5. Simplified 55.4%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-208}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \left(x \cdot \frac{y}{a\_m}\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* x (/ y a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (x * (y / a_m));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    code = a_s * (x * (y / a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (x * (y / a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	return a_s * (x * (y / a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(x * Float64(y / a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (x * (y / a_m));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in x around inf 51.6%

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

   1. associate-*r/ 53.4%

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

5. Simplified 53.4%

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

6. Final simplification 53.4%

   \[\leadsto x \cdot \frac{y}{a} \]
7. Add Preprocessing

Developer target: 91.4% 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 2024053 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (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))