Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Percentage Accurate: 91.2% → 93.7%

Time: 10.8s

Alternatives: 11

Speedup: 0.6×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+247}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (if (<= t_1 -2e+265)
     (* -4.5 (/ t (/ a z)))
     (if (<= t_1 5e+247)
       (+ (* -4.5 (/ (* z t) a)) (* 0.5 (/ (* x y) a)))
       (* z (* -4.5 (/ t a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = -4.5 * (t / (a / z));
	} else if (t_1 <= 5e+247) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else {
		tmp = z * (-4.5 * (t / a));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-2d+265)) then
        tmp = (-4.5d0) * (t / (a / z))
    else if (t_1 <= 5d+247) then
        tmp = ((-4.5d0) * ((z * t) / a)) + (0.5d0 * ((x * y) / a))
    else
        tmp = z * ((-4.5d0) * (t / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -2e+265) {
		tmp = -4.5 * (t / (a / z));
	} else if (t_1 <= 5e+247) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else {
		tmp = z * (-4.5 * (t / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -2e+265:
		tmp = -4.5 * (t / (a / z))
	elif t_1 <= 5e+247:
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a))
	else:
		tmp = z * (-4.5 * (t / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -2e+265)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif (t_1 <= 5e+247)
		tmp = Float64(Float64(-4.5 * Float64(Float64(z * t) / a)) + Float64(0.5 * Float64(Float64(x * y) / a)));
	else
		tmp = Float64(z * Float64(-4.5 * Float64(t / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -2e+265)
		tmp = -4.5 * (t / (a / z));
	elseif (t_1 <= 5e+247)
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	else
		tmp = z * (-4.5 * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+265], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+247], N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.5 * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z 9) t) < -2.00000000000000013e265

   1. Initial program 74.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 74.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 74.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if -2.00000000000000013e265 < (*.f64 (*.f64 z 9) t) < 5.00000000000000023e247

   1. Initial program 96.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 96.7%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 96.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   if 5.00000000000000023e247 < (*.f64 (*.f64 z 9) t)

   1. Initial program 76.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 76.7%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 72.7%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 76.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 76.8%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\left(\frac{t}{a} \cdot z\right)} \cdot -4.5 \]

      3. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right)} \cdot -4.5 \]

      4. associate-*l* 99.9%

         \[\leadsto \color{blue}{z \cdot \left(\frac{t}{a} \cdot -4.5\right)} \]

      5. *-commutative 99.9%

         \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]

      6. associate-*r/ 99.7%

         \[\leadsto z \cdot \color{blue}{\frac{-4.5 \cdot t}{a}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{z \cdot \frac{-4.5 \cdot t}{a}} \]

   8. Taylor expanded in t around 0 99.9%

      \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 5 \cdot 10^{+247}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 2: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \frac{z}{a} \cdot \left(\left(t \cdot 9\right) \cdot \left(-0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* a 2.0) 1e-32)
   (/ (fma x y (* -9.0 (* z t))) (* a 2.0))
   (fma x (* y (/ 0.5 a)) (* (/ z a) (* (* t 9.0) (- 0.5))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-32) {
		tmp = fma(x, y, (-9.0 * (z * t))) / (a * 2.0);
	} else {
		tmp = fma(x, (y * (0.5 / a)), ((z / a) * ((t * 9.0) * -0.5)));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 1e-32)
		tmp = Float64(fma(x, y, Float64(-9.0 * Float64(z * t))) / Float64(a * 2.0));
	else
		tmp = fma(x, Float64(y * Float64(0.5 / a)), Float64(Float64(z / a) * Float64(Float64(t * 9.0) * Float64(-0.5))));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 1e-32], N[(N[(x * y + N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(z / a), $MachinePrecision] * N[(N[(t * 9.0), $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a 2) < 1.00000000000000006e-32

   1. Initial program 95.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. fma-neg 95.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]

      2. *-commutative 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{\left(9 \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      3. associate-*l* 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{9 \cdot \left(z \cdot t\right)}\right)}{a \cdot 2} \]

      4. distribute-lft-neg-in 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(z \cdot t\right)}\right)}{a \cdot 2} \]

      5. metadata-eval 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(z \cdot t\right)\right)}{a \cdot 2} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}} \]

   if 1.00000000000000006e-32 < (*.f64 a 2)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 85.1%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. div-sub 85.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

      2. div-inv 85.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{a \cdot 2}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      3. *-commutative 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      4. associate-/r* 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      5. metadata-eval 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      6. times-frac 92.9%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

   5. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 92.9%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z}{a}\right) \cdot \frac{9 \cdot t}{2}} \]

      2. associate-*l* 93.5%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} + \left(-\frac{z}{a}\right) \cdot \frac{9 \cdot t}{2} \]

      3. fma-def 93.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \left(-\frac{z}{a}\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      4. div-inv 93.5%

         \[\leadsto \mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(\left(9 \cdot t\right) \cdot \frac{1}{2}\right)}\right) \]

      5. *-commutative 93.5%

         \[\leadsto \mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \left(-\frac{z}{a}\right) \cdot \left(\color{blue}{\left(t \cdot 9\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 93.5%

         \[\leadsto \mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \left(-\frac{z}{a}\right) \cdot \left(\left(t \cdot 9\right) \cdot \color{blue}{0.5}\right)\right) \]

   7. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \left(-\frac{z}{a}\right) \cdot \left(\left(t \cdot 9\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \frac{0.5}{a}, \frac{z}{a} \cdot \left(\left(t \cdot 9\right) \cdot \left(-0.5\right)\right)\right)\\ \end{array} \]

Alternative 3: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right) - \frac{z}{a} \cdot \frac{t \cdot 9}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* a 2.0) 1e-32)
   (/ (fma x y (* -9.0 (* z t))) (* a 2.0))
   (- (* (/ 0.5 a) (* x y)) (* (/ z a) (/ (* t 9.0) 2.0)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-32) {
		tmp = fma(x, y, (-9.0 * (z * t))) / (a * 2.0);
	} else {
		tmp = ((0.5 / a) * (x * y)) - ((z / a) * ((t * 9.0) / 2.0));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 1e-32)
		tmp = Float64(fma(x, y, Float64(-9.0 * Float64(z * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(x * y)) - Float64(Float64(z / a) * Float64(Float64(t * 9.0) / 2.0)));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 1e-32], N[(N[(x * y + N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(z / a), $MachinePrecision] * N[(N[(t * 9.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a 2) < 1.00000000000000006e-32

   1. Initial program 95.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. fma-neg 95.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]

      2. *-commutative 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{\left(9 \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      3. associate-*l* 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{9 \cdot \left(z \cdot t\right)}\right)}{a \cdot 2} \]

      4. distribute-lft-neg-in 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(z \cdot t\right)}\right)}{a \cdot 2} \]

      5. metadata-eval 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(z \cdot t\right)\right)}{a \cdot 2} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}} \]

   if 1.00000000000000006e-32 < (*.f64 a 2)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 85.1%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. div-sub 85.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

      2. div-inv 85.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{a \cdot 2}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      3. *-commutative 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      4. associate-/r* 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      5. metadata-eval 85.0%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      6. times-frac 92.9%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

   5. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -9 \cdot \left(z \cdot t\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right) - \frac{z}{a} \cdot \frac{t \cdot 9}{2}\\ \end{array} \]

Alternative 4: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+217}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(t \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (if (<= t_1 -4e+217)
     (* -4.5 (/ t (/ a z)))
     (if (<= t_1 5e+205)
       (/ (- (* x y) (* z (* t 9.0))) (* a 2.0))
       (* z (* -4.5 (/ t a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -4e+217) {
		tmp = -4.5 * (t / (a / z));
	} else if (t_1 <= 5e+205) {
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	} else {
		tmp = z * (-4.5 * (t / a));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-4d+217)) then
        tmp = (-4.5d0) * (t / (a / z))
    else if (t_1 <= 5d+205) then
        tmp = ((x * y) - (z * (t * 9.0d0))) / (a * 2.0d0)
    else
        tmp = z * ((-4.5d0) * (t / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -4e+217) {
		tmp = -4.5 * (t / (a / z));
	} else if (t_1 <= 5e+205) {
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	} else {
		tmp = z * (-4.5 * (t / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -4e+217:
		tmp = -4.5 * (t / (a / z))
	elif t_1 <= 5e+205:
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0)
	else:
		tmp = z * (-4.5 * (t / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -4e+217)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif (t_1 <= 5e+205)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(t * 9.0))) / Float64(a * 2.0));
	else
		tmp = Float64(z * Float64(-4.5 * Float64(t / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -4e+217)
		tmp = -4.5 * (t / (a / z));
	elseif (t_1 <= 5e+205)
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	else
		tmp = z * (-4.5 * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+217], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+205], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(t * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.5 * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z 9) t) < -3.99999999999999984e217

   1. Initial program 77.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 77.0%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 77.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 77.0%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if -3.99999999999999984e217 < (*.f64 (*.f64 z 9) t) < 5.0000000000000002e205

   1. Initial program 97.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 97.1%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   if 5.0000000000000002e205 < (*.f64 (*.f64 z 9) t)

   1. Initial program 78.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 78.1%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 81.5%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\left(\frac{t}{a} \cdot z\right)} \cdot -4.5 \]

      3. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right)} \cdot -4.5 \]

      4. associate-*l* 99.8%

         \[\leadsto \color{blue}{z \cdot \left(\frac{t}{a} \cdot -4.5\right)} \]

      5. *-commutative 99.8%

         \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]

      6. associate-*r/ 99.7%

         \[\leadsto z \cdot \color{blue}{\frac{-4.5 \cdot t}{a}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{z \cdot \frac{-4.5 \cdot t}{a}} \]

   8. Taylor expanded in t around 0 99.8%

      \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -4 \cdot 10^{+217}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(t \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 5: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-119}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (* y (/ 0.5 a)))))
   (if (<= (* x y) -2e+40)
     t_1
     (if (<= (* x y) 4e-119)
       (* -4.5 (/ t (/ a z)))
       (if (<= (* x y) 2e-52)
         t_1
         (if (<= (* x y) 2e+26)
           (* -4.5 (/ (* z t) a))
           (* y (* 0.5 (/ x a)))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y * (0.5 / a));
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = t_1;
	} else if ((x * y) <= 4e-119) {
		tmp = -4.5 * (t / (a / z));
	} else if ((x * y) <= 2e-52) {
		tmp = t_1;
	} else if ((x * y) <= 2e+26) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y * (0.5 * (x / a));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * (0.5d0 / a))
    if ((x * y) <= (-2d+40)) then
        tmp = t_1
    else if ((x * y) <= 4d-119) then
        tmp = (-4.5d0) * (t / (a / z))
    else if ((x * y) <= 2d-52) then
        tmp = t_1
    else if ((x * y) <= 2d+26) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = y * (0.5d0 * (x / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y * (0.5 / a));
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = t_1;
	} else if ((x * y) <= 4e-119) {
		tmp = -4.5 * (t / (a / z));
	} else if ((x * y) <= 2e-52) {
		tmp = t_1;
	} else if ((x * y) <= 2e+26) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y * (0.5 * (x / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = x * (y * (0.5 / a))
	tmp = 0
	if (x * y) <= -2e+40:
		tmp = t_1
	elif (x * y) <= 4e-119:
		tmp = -4.5 * (t / (a / z))
	elif (x * y) <= 2e-52:
		tmp = t_1
	elif (x * y) <= 2e+26:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = y * (0.5 * (x / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y * Float64(0.5 / a)))
	tmp = 0.0
	if (Float64(x * y) <= -2e+40)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-119)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif (Float64(x * y) <= 2e-52)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+26)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(y * Float64(0.5 * Float64(x / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y * (0.5 / a));
	tmp = 0.0;
	if ((x * y) <= -2e+40)
		tmp = t_1;
	elseif ((x * y) <= 4e-119)
		tmp = -4.5 * (t / (a / z));
	elseif ((x * y) <= 2e-52)
		tmp = t_1;
	elseif ((x * y) <= 2e+26)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = y * (0.5 * (x / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+40], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-119], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-52], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+26], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.00000000000000006e40 or 4.00000000000000005e-119 < (*.f64 x y) < 2e-52

   1. Initial program 93.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 93.6%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 93.7%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 56.9%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-udef 40.2%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]

   6. Applied egg-rr 40.2%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-log1p 93.7%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      3. associate-/l* 93.2%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   8. Simplified 93.2%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   10. Applied egg-rr 91.3%

       \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{a}{y}}\right)} \]

   11. Taylor expanded in t around 0 78.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 78.8%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{a}} \]

       2. associate-*l/ 78.7%

          \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right)} \]

       3. *-commutative 78.7%

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a}} \]

       4. associate-*l* 79.2%

          \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   13. Simplified 79.2%

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   if -2.00000000000000006e40 < (*.f64 x y) < 4.00000000000000005e-119

   1. Initial program 91.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 91.5%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 92.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 77.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 79.6%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 79.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if 2e-52 < (*.f64 x y) < 2.0000000000000001e26

   1. Initial program 99.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 99.6%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 60.5%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 2.0000000000000001e26 < (*.f64 x y)

   1. Initial program 91.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 91.5%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 88.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 49.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-udef 44.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]

   6. Applied egg-rr 44.6%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 49.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-log1p 88.1%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      3. associate-/l* 89.7%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   8. Simplified 89.7%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   9. Taylor expanded in t around 0 82.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   10. Step-by-step derivation

       1. associate-*l/ 83.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{a} \cdot y\right)} \]

       2. *-commutative 83.0%

          \[\leadsto \color{blue}{\left(\frac{x}{a} \cdot y\right) \cdot 0.5} \]

       3. *-commutative 83.0%

          \[\leadsto \color{blue}{\left(y \cdot \frac{x}{a}\right)} \cdot 0.5 \]

       4. associate-*l* 83.0%

          \[\leadsto \color{blue}{y \cdot \left(\frac{x}{a} \cdot 0.5\right)} \]

   11. Simplified 83.0%

       \[\leadsto \color{blue}{y \cdot \left(\frac{x}{a} \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-119}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 6: 71.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-119}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (* y (/ 0.5 a)))))
   (if (<= (* x y) -2e+40)
     t_1
     (if (<= (* x y) 4e-119)
       (* z (* -4.5 (/ t a)))
       (if (<= (* x y) 2e-52)
         t_1
         (if (<= (* x y) 2e+26)
           (* -4.5 (/ (* z t) a))
           (* y (* 0.5 (/ x a)))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y * (0.5 / a));
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = t_1;
	} else if ((x * y) <= 4e-119) {
		tmp = z * (-4.5 * (t / a));
	} else if ((x * y) <= 2e-52) {
		tmp = t_1;
	} else if ((x * y) <= 2e+26) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y * (0.5 * (x / a));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * (0.5d0 / a))
    if ((x * y) <= (-2d+40)) then
        tmp = t_1
    else if ((x * y) <= 4d-119) then
        tmp = z * ((-4.5d0) * (t / a))
    else if ((x * y) <= 2d-52) then
        tmp = t_1
    else if ((x * y) <= 2d+26) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = y * (0.5d0 * (x / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y * (0.5 / a));
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = t_1;
	} else if ((x * y) <= 4e-119) {
		tmp = z * (-4.5 * (t / a));
	} else if ((x * y) <= 2e-52) {
		tmp = t_1;
	} else if ((x * y) <= 2e+26) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = y * (0.5 * (x / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = x * (y * (0.5 / a))
	tmp = 0
	if (x * y) <= -2e+40:
		tmp = t_1
	elif (x * y) <= 4e-119:
		tmp = z * (-4.5 * (t / a))
	elif (x * y) <= 2e-52:
		tmp = t_1
	elif (x * y) <= 2e+26:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = y * (0.5 * (x / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y * Float64(0.5 / a)))
	tmp = 0.0
	if (Float64(x * y) <= -2e+40)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-119)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a)));
	elseif (Float64(x * y) <= 2e-52)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+26)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(y * Float64(0.5 * Float64(x / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y * (0.5 / a));
	tmp = 0.0;
	if ((x * y) <= -2e+40)
		tmp = t_1;
	elseif ((x * y) <= 4e-119)
		tmp = z * (-4.5 * (t / a));
	elseif ((x * y) <= 2e-52)
		tmp = t_1;
	elseif ((x * y) <= 2e+26)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = y * (0.5 * (x / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+40], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-119], N[(z * N[(-4.5 * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-52], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+26], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.00000000000000006e40 or 4.00000000000000005e-119 < (*.f64 x y) < 2e-52

   1. Initial program 93.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 93.6%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 93.7%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 56.9%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-udef 40.2%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]

   6. Applied egg-rr 40.2%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-log1p 93.7%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      3. associate-/l* 93.2%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   8. Simplified 93.2%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   10. Applied egg-rr 91.3%

       \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{a}{y}}\right)} \]

   11. Taylor expanded in t around 0 78.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 78.8%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{a}} \]

       2. associate-*l/ 78.7%

          \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right)} \]

       3. *-commutative 78.7%

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a}} \]

       4. associate-*l* 79.2%

          \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   13. Simplified 79.2%

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   if -2.00000000000000006e40 < (*.f64 x y) < 4.00000000000000005e-119

   1. Initial program 91.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 91.5%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 92.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 77.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

      2. associate-*l/ 83.8%

         \[\leadsto \color{blue}{\left(\frac{t}{a} \cdot z\right)} \cdot -4.5 \]

      3. *-commutative 83.8%

         \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right)} \cdot -4.5 \]

      4. associate-*l* 84.0%

         \[\leadsto \color{blue}{z \cdot \left(\frac{t}{a} \cdot -4.5\right)} \]

      5. *-commutative 84.0%

         \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]

      6. associate-*r/ 83.9%

         \[\leadsto z \cdot \color{blue}{\frac{-4.5 \cdot t}{a}} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{z \cdot \frac{-4.5 \cdot t}{a}} \]

   8. Taylor expanded in t around 0 84.0%

      \[\leadsto z \cdot \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \]

   if 2e-52 < (*.f64 x y) < 2.0000000000000001e26

   1. Initial program 99.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 99.6%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 60.5%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 2.0000000000000001e26 < (*.f64 x y)

   1. Initial program 91.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 91.5%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 88.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 49.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-udef 44.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]

   6. Applied egg-rr 44.6%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 49.6%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-log1p 88.1%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      3. associate-/l* 89.7%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   8. Simplified 89.7%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   9. Taylor expanded in t around 0 82.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   10. Step-by-step derivation

       1. associate-*l/ 83.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{a} \cdot y\right)} \]

       2. *-commutative 83.0%

          \[\leadsto \color{blue}{\left(\frac{x}{a} \cdot y\right) \cdot 0.5} \]

       3. *-commutative 83.0%

          \[\leadsto \color{blue}{\left(y \cdot \frac{x}{a}\right)} \cdot 0.5 \]

       4. associate-*l* 83.0%

          \[\leadsto \color{blue}{y \cdot \left(\frac{x}{a} \cdot 0.5\right)} \]

   11. Simplified 83.0%

       \[\leadsto \color{blue}{y \cdot \left(\frac{x}{a} \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-119}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 7: 91.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(t \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right) - \frac{z}{a} \cdot \frac{t \cdot 9}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* a 2.0) 2e-53)
   (/ (- (* x y) (* z (* t 9.0))) (* a 2.0))
   (- (* (/ 0.5 a) (* x y)) (* (/ z a) (/ (* t 9.0) 2.0)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-53) {
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	} else {
		tmp = ((0.5 / a) * (x * y)) - ((z / a) * ((t * 9.0) / 2.0));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a * 2.0d0) <= 2d-53) then
        tmp = ((x * y) - (z * (t * 9.0d0))) / (a * 2.0d0)
    else
        tmp = ((0.5d0 / a) * (x * y)) - ((z / a) * ((t * 9.0d0) / 2.0d0))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-53) {
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	} else {
		tmp = ((0.5 / a) * (x * y)) - ((z / a) * ((t * 9.0) / 2.0));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (a * 2.0) <= 2e-53:
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0)
	else:
		tmp = ((0.5 / a) * (x * y)) - ((z / a) * ((t * 9.0) / 2.0))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 2e-53)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(t * 9.0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(x * y)) - Float64(Float64(z / a) * Float64(Float64(t * 9.0) / 2.0)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a * 2.0) <= 2e-53)
		tmp = ((x * y) - (z * (t * 9.0))) / (a * 2.0);
	else
		tmp = ((0.5 / a) * (x * y)) - ((z / a) * ((t * 9.0) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 2e-53], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(t * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(z / a), $MachinePrecision] * N[(N[(t * 9.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a 2) < 2.00000000000000006e-53

   1. Initial program 95.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 95.8%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   if 2.00000000000000006e-53 < (*.f64 a 2)

   1. Initial program 85.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 85.4%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. div-sub 85.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

      2. div-inv 85.4%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{a \cdot 2}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      3. *-commutative 85.4%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      4. associate-/r* 85.4%

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      5. metadata-eval 85.4%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} - \frac{z \cdot \left(9 \cdot t\right)}{a \cdot 2} \]

      6. times-frac 93.0%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

   5. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(t \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right) - \frac{z}{a} \cdot \frac{t \cdot 9}{2}\\ \end{array} \]

Alternative 8: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-126}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+40)
   (* 0.5 (/ x (/ a y)))
   (if (<= (* x y) 5e-126) (* -4.5 (/ t (/ a z))) (* 0.5 (/ (* x y) a)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = 0.5 * (x / (a / y));
	} else if ((x * y) <= 5e-126) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = 0.5 * ((x * y) / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+40)) then
        tmp = 0.5d0 * (x / (a / y))
    else if ((x * y) <= 5d-126) then
        tmp = (-4.5d0) * (t / (a / z))
    else
        tmp = 0.5d0 * ((x * y) / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = 0.5 * (x / (a / y));
	} else if ((x * y) <= 5e-126) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = 0.5 * ((x * y) / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+40:
		tmp = 0.5 * (x / (a / y))
	elif (x * y) <= 5e-126:
		tmp = -4.5 * (t / (a / z))
	else:
		tmp = 0.5 * ((x * y) / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+40)
		tmp = Float64(0.5 * Float64(x / Float64(a / y)));
	elseif (Float64(x * y) <= 5e-126)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	else
		tmp = Float64(0.5 * Float64(Float64(x * y) / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+40)
		tmp = 0.5 * (x / (a / y));
	elseif ((x * y) <= 5e-126)
		tmp = -4.5 * (t / (a / z));
	else
		tmp = 0.5 * ((x * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+40], N[(0.5 * N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-126], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000006e40

   1. Initial program 92.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 92.2%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around inf 79.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* 83.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Simplified 83.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\frac{a}{y}}} \]

   if -2.00000000000000006e40 < (*.f64 x y) < 5.00000000000000006e-126

   1. Initial program 92.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 92.2%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 93.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 78.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 80.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if 5.00000000000000006e-126 < (*.f64 x y)

   1. Initial program 93.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 93.3%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around inf 73.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-126}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \]

Alternative 9: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-126}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+40)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 5e-126) (* -4.5 (/ t (/ a z))) (* 0.5 (/ (* x y) a)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 5e-126) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = 0.5 * ((x * y) / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+40)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 5d-126) then
        tmp = (-4.5d0) * (t / (a / z))
    else
        tmp = 0.5d0 * ((x * y) / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+40) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 5e-126) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = 0.5 * ((x * y) / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+40:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 5e-126:
		tmp = -4.5 * (t / (a / z))
	else:
		tmp = 0.5 * ((x * y) / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+40)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 5e-126)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	else
		tmp = Float64(0.5 * Float64(Float64(x * y) / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+40)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 5e-126)
		tmp = -4.5 * (t / (a / z));
	else
		tmp = 0.5 * ((x * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+40], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-126], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000006e40

   1. Initial program 92.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 92.2%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 92.2%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 52.8%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-udef 42.7%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]

   6. Applied egg-rr 42.7%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 52.8%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. expm1-log1p 92.2%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x \cdot y}{a}} \]

      3. associate-/l* 94.0%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   8. Simplified 94.0%

      \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   10. Applied egg-rr 92.4%

       \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{a}{y}}\right)} \]

   11. Taylor expanded in t around 0 79.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 79.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{a}} \]

       2. associate-*l/ 79.4%

          \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right)} \]

       3. *-commutative 79.4%

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a}} \]

       4. associate-*l* 81.3%

          \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   13. Simplified 81.3%

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} \]

   if -2.00000000000000006e40 < (*.f64 x y) < 5.00000000000000006e-126

   1. Initial program 92.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 92.2%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 93.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 78.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 80.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if 5.00000000000000006e-126 < (*.f64 x y)

   1. Initial program 93.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 93.3%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around inf 73.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-126}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \]

Alternative 10: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+78}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.6e+78)
   (* -4.5 (/ t (/ a z)))
   (if (<= z 3.5e-146) (* 0.5 (/ x (/ a y))) (* -4.5 (* z (/ t a))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+78) {
		tmp = -4.5 * (t / (a / z));
	} else if (z <= 3.5e-146) {
		tmp = 0.5 * (x / (a / y));
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.6d+78)) then
        tmp = (-4.5d0) * (t / (a / z))
    else if (z <= 3.5d-146) then
        tmp = 0.5d0 * (x / (a / y))
    else
        tmp = (-4.5d0) * (z * (t / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+78) {
		tmp = -4.5 * (t / (a / z));
	} else if (z <= 3.5e-146) {
		tmp = 0.5 * (x / (a / y));
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.6e+78:
		tmp = -4.5 * (t / (a / z))
	elif z <= 3.5e-146:
		tmp = 0.5 * (x / (a / y))
	else:
		tmp = -4.5 * (z * (t / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.6e+78)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif (z <= 3.5e-146)
		tmp = Float64(0.5 * Float64(x / Float64(a / y)));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.6e+78)
		tmp = -4.5 * (t / (a / z));
	elseif (z <= 3.5e-146)
		tmp = 0.5 * (x / (a / y));
	else
		tmp = -4.5 * (z * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.6e+78], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e-146], N[(0.5 * N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.6e78

   1. Initial program 83.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 83.2%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 83.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{x \cdot y}{a}} \]

   5. Taylor expanded in t around inf 61.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. associate-/l* 69.1%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   7. Simplified 69.1%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]

   if -6.6e78 < z < 3.5000000000000001e-146

   1. Initial program 96.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 96.4%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around inf 75.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* 69.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]

   6. Simplified 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\frac{a}{y}}} \]

   if 3.5000000000000001e-146 < z

   1. Initial program 92.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. associate-*l* 92.3%

         \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

   4. Taylor expanded in x around 0 59.7%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* 65.9%

         \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

      2. associate-/r/ 67.0%

         \[\leadsto -4.5 \cdot \color{blue}{\left(\frac{t}{a} \cdot z\right)} \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{-4.5 \cdot \left(\frac{t}{a} \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+78}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 11: 51.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ -4.5 \cdot \left(z \cdot \frac{t}{a}\right) \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* -4.5 (* z (/ t a))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return -4.5 * (z * (t / a));
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (-4.5d0) * (z * (t / a))
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	return -4.5 * (z * (t / a));
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return -4.5 * (z * (t / a))

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(-4.5 * Float64(z * Float64(t / a)))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = -4.5 * (z * (t / a));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Step-by-step derivation

   1. associate-*l* 92.6%

      \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

3. Simplified 92.6%

   \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]

4. Taylor expanded in x around 0 50.1%

   \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
5. Step-by-step derivation

   1. associate-/l* 52.0%

      \[\leadsto -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]

   2. associate-/r/ 54.2%

      \[\leadsto -4.5 \cdot \color{blue}{\left(\frac{t}{a} \cdot z\right)} \]

6. Simplified 54.2%

   \[\leadsto \color{blue}{-4.5 \cdot \left(\frac{t}{a} \cdot z\right)} \]

7. Final simplification 54.2%

   \[\leadsto -4.5 \cdot \left(z \cdot \frac{t}{a}\right) \]

Developer target: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	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 (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023285 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))