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

Percentage Accurate: 91.2% → 97.3%

Time: 8.0s

Alternatives: 7

Speedup: 0.6×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+264} \lor \neg \left(t_1 \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{0.5}{a}, t \cdot -9, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -1e+264) (not (<= t_1 2e+301)))
     (fma (* z (/ 0.5 a)) (* t -9.0) (* 0.5 (/ y (/ a x))))
     (/ t_1 (* a 2.0)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -1e+264) || !(t_1 <= 2e+301)) {
		tmp = fma((z * (0.5 / a)), (t * -9.0), (0.5 * (y / (a / x))));
	} else {
		tmp = t_1 / (a * 2.0);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -1e+264) || !(t_1 <= 2e+301))
		tmp = fma(Float64(z * Float64(0.5 / a)), Float64(t * -9.0), Float64(0.5 * Float64(y / Float64(a / x))));
	else
		tmp = Float64(t_1 / Float64(a * 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_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+264], N[Not[LessEqual[t$95$1, 2e+301]], $MachinePrecision]], N[(N[(z * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * N[(t * -9.0), $MachinePrecision] + N[(0.5 * N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1.00000000000000004e264 or 2.00000000000000011e301 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 67.2%

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

      1. associate-*l* 68.5%

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

   3. Simplified 68.5%

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

      1. sub-neg 68.5%

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

      2. +-commutative 68.5%

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

      3. distribute-rgt-neg-in 68.5%

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

      4. distribute-lft-neg-in 68.5%

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

      5. metadata-eval 68.5%

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

      6. *-commutative 68.5%

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

      7. associate-*r* 68.4%

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

      8. fma-def 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. div-inv 68.4%

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

      2. metadata-eval 68.4%

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

      3. div-inv 68.4%

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

      4. clear-num 68.4%

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

      5. *-commutative 68.4%

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

      6. *-commutative 68.4%

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

      7. fma-def 68.4%

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

      8. *-commutative 68.4%

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

      9. +-commutative 68.4%

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

      10. *-commutative 68.4%

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

      11. +-commutative 68.4%

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

      12. distribute-lft-in 64.2%

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

      13. *-commutative 64.2%

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

      14. associate-*r* 64.3%

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

      15. *-commutative 64.3%

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

      16. *-commutative 64.3%

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

      17. associate-*r* 75.9%

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

      18. fma-def 75.9%

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

      19. associate-*l/ 75.9%

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

   7. Applied egg-rr 92.9%

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

   if -1.00000000000000004e264 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 2.00000000000000011e301

   1. Initial program 98.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+264} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{0.5}{a}, t \cdot -9, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \end{array} \]

Alternative 2: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \end{array} \end{array} \]

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) (- INFINITY))
   (* 0.5 (* y (/ x a)))
   (if (<= (* x y) 5e+260)
     (/ (fma (* z t) -9.0 (* x y)) (* a 2.0))
     (/ (* y 0.5) (/ a x)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = 0.5 * (y * (x / a));
	} else if ((x * y) <= 5e+260) {
		tmp = fma((z * t), -9.0, (x * y)) / (a * 2.0);
	} else {
		tmp = (y * 0.5) / (a / x);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	elseif (Float64(x * y) <= 5e+260)
		tmp = Float64(fma(Float64(z * t), -9.0, Float64(x * y)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(y * 0.5) / Float64(a / x));
	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[(x * y), $MachinePrecision], (-Infinity)], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+260], N[(N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / N[(a / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 68.4%

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

      1. sub-neg 68.4%

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

      2. +-commutative 68.4%

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

      3. neg-sub0 68.4%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 68.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 68.4%

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

      6. neg-mul-1 68.4%

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

      7. associate-/l* 68.4%

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

      8. associate-/r/ 68.4%

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

      9. *-commutative 68.4%

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

      10. sub-neg 68.4%

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

      11. +-commutative 68.4%

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

      12. neg-sub0 68.4%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 68.4%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 68.4%

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

      15. distribute-lft-neg-out 68.4%

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

      16. distribute-rgt-neg-in 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in x around inf 68.4%

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

      1. expm1-log1p-u 28.4%

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

      2. expm1-udef 28.4%

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

      3. associate-/l* 37.6%

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

   6. Applied egg-rr 37.6%

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

      1. expm1-def 37.6%

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

      2. expm1-log1p 99.8%

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

      3. associate-/l* 68.4%

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

      4. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 x y) < 4.9999999999999996e260

   1. Initial program 93.8%

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

      1. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

      1. sub-neg 93.9%

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

      2. +-commutative 93.9%

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

      3. distribute-rgt-neg-in 93.9%

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

      4. distribute-lft-neg-in 93.9%

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

      5. metadata-eval 93.9%

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

      6. *-commutative 93.9%

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

      7. associate-*r* 94.3%

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

      8. fma-def 94.3%

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

   5. Applied egg-rr 94.3%

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

   if 4.9999999999999996e260 < (*.f64 x y)

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

   3. Simplified 63.0%

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

      1. sub-neg 63.0%

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

      2. +-commutative 63.0%

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

      3. distribute-rgt-neg-in 63.0%

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

      4. distribute-lft-neg-in 63.0%

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

      5. metadata-eval 63.0%

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

      6. *-commutative 63.0%

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

      7. associate-*r* 63.0%

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

      8. fma-def 63.0%

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

   5. Applied egg-rr 63.0%

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

   6. Taylor expanded in z around 0 63.0%

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

      1. associate-/l* 99.8%

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

      2. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \end{array} \]

Alternative 3: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq 10^{+214}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + -9 \cdot \left(z \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* x y) (- INFINITY)) (not (<= (* x y) 1e+214)))
   (* 0.5 (* y (/ x a)))
   (* (/ 0.5 a) (+ (* x y) (* -9.0 (* z t))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -((double) INFINITY)) || !((x * y) <= 1e+214)) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = (0.5 / a) * ((x * y) + (-9.0 * (z * t)));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -Double.POSITIVE_INFINITY) || !((x * y) <= 1e+214)) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = (0.5 / a) * ((x * y) + (-9.0 * (z * t)));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) <= -math.inf) or not ((x * y) <= 1e+214):
		tmp = 0.5 * (y * (x / a))
	else:
		tmp = (0.5 / a) * ((x * y) + (-9.0 * (z * t)))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(x * y) <= Float64(-Inf)) || !(Float64(x * y) <= 1e+214))
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(x * y) + Float64(-9.0 * Float64(z * t))));
	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) <= -Inf) || ~(((x * y) <= 1e+214)))
		tmp = 0.5 * (y * (x / a));
	else
		tmp = (0.5 / a) * ((x * y) + (-9.0 * (z * t)));
	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[Or[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+214]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] + N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 9.9999999999999995e213 < (*.f64 x y)

   1. Initial program 72.0%

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

      1. sub-neg 72.0%

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

      2. +-commutative 72.0%

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

      3. neg-sub0 72.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 72.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 72.0%

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

      6. neg-mul-1 72.0%

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

      7. associate-/l* 71.9%

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

      8. associate-/r/ 71.9%

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

      9. *-commutative 71.9%

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

      10. sub-neg 71.9%

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

      11. +-commutative 71.9%

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

      12. neg-sub0 71.9%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 71.9%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 71.9%

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

      15. distribute-lft-neg-out 71.9%

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

      16. distribute-rgt-neg-in 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 72.0%

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

      1. expm1-log1p-u 34.0%

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

      2. expm1-udef 31.9%

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

      3. associate-/l* 41.5%

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

   6. Applied egg-rr 41.5%

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

      1. expm1-def 43.7%

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

      2. expm1-log1p 99.8%

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

      3. associate-/l* 72.0%

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

      4. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   if -inf.0 < (*.f64 x y) < 9.9999999999999995e213

   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. sub-neg 93.6%

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

      2. +-commutative 93.6%

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

      3. neg-sub0 93.6%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 93.6%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 93.6%

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

      6. neg-mul-1 93.6%

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

      7. associate-/l* 93.6%

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

      8. associate-/r/ 93.5%

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

      9. *-commutative 93.5%

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

      10. sub-neg 93.5%

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

      11. +-commutative 93.5%

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

      12. neg-sub0 93.5%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 93.5%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 93.5%

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

      15. distribute-lft-neg-out 93.5%

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

      16. distribute-rgt-neg-in 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 93.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq 10^{+214}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + -9 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]

Alternative 4: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -4e+123)
   (* 0.5 (* y (/ x a)))
   (if (<= (* x y) 5e+29) (* -4.5 (/ t (/ a z))) (* (/ y a) (* x 0.5)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -4e+123) {
		tmp = 0.5 * (y * (x / a));
	} else if ((x * y) <= 5e+29) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = (y / a) * (x * 0.5);
	}
	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) <= (-4d+123)) then
        tmp = 0.5d0 * (y * (x / a))
    else if ((x * y) <= 5d+29) then
        tmp = (-4.5d0) * (t / (a / z))
    else
        tmp = (y / a) * (x * 0.5d0)
    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) <= -4e+123) {
		tmp = 0.5 * (y * (x / a));
	} else if ((x * y) <= 5e+29) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = (y / a) * (x * 0.5);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -4e+123:
		tmp = 0.5 * (y * (x / a))
	elif (x * y) <= 5e+29:
		tmp = -4.5 * (t / (a / z))
	else:
		tmp = (y / a) * (x * 0.5)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -4e+123)
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	elseif (Float64(x * y) <= 5e+29)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	else
		tmp = Float64(Float64(y / a) * Float64(x * 0.5));
	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) <= -4e+123)
		tmp = 0.5 * (y * (x / a));
	elseif ((x * y) <= 5e+29)
		tmp = -4.5 * (t / (a / z));
	else
		tmp = (y / a) * (x * 0.5);
	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], -4e+123], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+29], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999991e123

   1. Initial program 85.2%

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

      1. sub-neg 85.2%

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

      2. +-commutative 85.2%

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

      3. neg-sub0 85.2%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 85.2%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 85.2%

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

      6. neg-mul-1 85.2%

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

      7. associate-/l* 85.2%

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

      8. associate-/r/ 85.1%

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

      9. *-commutative 85.1%

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

      10. sub-neg 85.1%

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

      11. +-commutative 85.1%

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

      12. neg-sub0 85.1%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 85.1%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 85.1%

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

      15. distribute-lft-neg-out 85.1%

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

      16. distribute-rgt-neg-in 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in x around inf 78.7%

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

      1. expm1-log1p-u 28.9%

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

      2. expm1-udef 22.5%

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

      3. associate-/l* 26.2%

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

   6. Applied egg-rr 26.2%

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

      1. expm1-def 30.5%

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

      2. expm1-log1p 88.2%

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

      3. associate-/l* 78.7%

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

      4. associate-*r/ 87.1%

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

   8. Simplified 87.1%

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

   if -3.99999999999999991e123 < (*.f64 x y) < 5.0000000000000001e29

   1. Initial program 92.8%

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

      1. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

      1. sub-neg 92.8%

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

      2. +-commutative 92.8%

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

      3. distribute-rgt-neg-in 92.8%

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

      4. distribute-lft-neg-in 92.8%

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

      5. metadata-eval 92.8%

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

      6. *-commutative 92.8%

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

      7. associate-*r* 93.3%

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

      8. fma-def 93.3%

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

   5. Applied egg-rr 93.3%

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

   6. Taylor expanded in z around inf 74.3%

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

      1. associate-/l* 71.9%

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

   8. Simplified 71.9%

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

   if 5.0000000000000001e29 < (*.f64 x y)

   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* 85.0%

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

   3. Simplified 85.0%

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

      1. sub-neg 85.0%

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

      2. +-commutative 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. distribute-lft-neg-in 85.0%

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

      5. metadata-eval 85.0%

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

      6. *-commutative 85.0%

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

      7. associate-*r* 85.0%

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

      8. fma-def 85.0%

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

   5. Applied egg-rr 85.0%

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

   6. Taylor expanded in z around 0 77.7%

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

      1. associate-/l* 85.1%

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

      2. associate-*r/ 85.1%

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

   8. Simplified 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 85.1%

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

      3. associate-/r/ 82.1%

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

      4. associate-*l* 82.1%

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-42} \lor \neg \left(y \leq 2.6 \cdot 10^{+69}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -2.2e-42) (not (<= y 2.6e+69)))
   (* 0.5 (* y (/ x a)))
   (* -4.5 (/ t (/ a z)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -2.2e-42) || !(y <= 2.6e+69)) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = -4.5 * (t / (a / z));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y <= (-2.2d-42)) .or. (.not. (y <= 2.6d+69))) then
        tmp = 0.5d0 * (y * (x / a))
    else
        tmp = (-4.5d0) * (t / (a / z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -2.2e-42) || !(y <= 2.6e+69)) {
		tmp = 0.5 * (y * (x / a));
	} else {
		tmp = -4.5 * (t / (a / z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (y <= -2.2e-42) or not (y <= 2.6e+69):
		tmp = 0.5 * (y * (x / a))
	else:
		tmp = -4.5 * (t / (a / z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -2.2e-42) || !(y <= 2.6e+69))
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	else
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -2.2e-42) || ~((y <= 2.6e+69)))
		tmp = 0.5 * (y * (x / a));
	else
		tmp = -4.5 * (t / (a / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -2.2e-42], N[Not[LessEqual[y, 2.6e+69]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000005e-42 or 2.6000000000000002e69 < y

   1. Initial program 86.0%

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

      1. sub-neg 86.0%

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

      2. +-commutative 86.0%

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

      3. neg-sub0 86.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 86.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 86.0%

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

      6. neg-mul-1 86.0%

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

      7. associate-/l* 86.0%

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

      8. associate-/r/ 85.9%

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

      9. *-commutative 85.9%

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

      10. sub-neg 85.9%

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

      11. +-commutative 85.9%

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

      12. neg-sub0 85.9%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 85.9%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 85.9%

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

      15. distribute-lft-neg-out 85.9%

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

      16. distribute-rgt-neg-in 85.9%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in x around inf 62.5%

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

      1. expm1-log1p-u 34.5%

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

      2. expm1-udef 26.1%

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

      3. associate-/l* 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. expm1-def 35.6%

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

      2. expm1-log1p 69.7%

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

      3. associate-/l* 62.5%

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

      4. associate-*r/ 70.4%

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

   8. Simplified 70.4%

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

   if -2.20000000000000005e-42 < y < 2.6000000000000002e69

   1. Initial program 93.2%

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

      1. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

      1. sub-neg 93.9%

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

      2. +-commutative 93.9%

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

      3. distribute-rgt-neg-in 93.9%

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

      4. distribute-lft-neg-in 93.9%

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

      5. metadata-eval 93.9%

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

      6. *-commutative 93.9%

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

      7. associate-*r* 93.8%

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

      8. fma-def 93.8%

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

   5. Applied egg-rr 93.8%

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

   6. Taylor expanded in z around inf 70.7%

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

      1. associate-/l* 69.4%

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

   8. Simplified 69.4%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t}{\frac{a}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-42} \lor \neg \left(y \leq 2.6 \cdot 10^{+69}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 6: 51.4% 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 89.9%

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

   1. sub-neg 89.9%

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

   2. +-commutative 89.9%

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

   3. neg-sub0 89.9%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

   4. associate-+l- 89.9%

      \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

   5. sub0-neg 89.9%

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

   6. neg-mul-1 89.9%

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

   7. associate-/l* 89.8%

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

   8. associate-/r/ 89.8%

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

   9. *-commutative 89.8%

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

   10. sub-neg 89.8%

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

   11. +-commutative 89.8%

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

   12. neg-sub0 89.8%

       \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

   13. associate-+l- 89.8%

       \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

   14. sub0-neg 89.8%

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

   15. distribute-lft-neg-out 89.8%

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

   16. distribute-rgt-neg-in 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in x around 0 52.4%

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

   1. associate-/l* 52.0%

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

   2. associate-/r/ 54.0%

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

6. Simplified 54.0%

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

7. Final simplification 54.0%

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

Alternative 7: 51.7% accurate, 1.9× speedup

Math

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

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 (/ t (/ a z))))

C

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

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) * (t / (a / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 89.9%

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

   1. associate-*l* 89.9%

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

3. Simplified 89.9%

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

   1. sub-neg 89.9%

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

   2. +-commutative 89.9%

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

   3. distribute-rgt-neg-in 89.9%

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

   4. distribute-lft-neg-in 89.9%

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

   5. metadata-eval 89.9%

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

   6. *-commutative 89.9%

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

   7. associate-*r* 90.3%

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

   8. fma-def 90.3%

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

5. Applied egg-rr 90.3%

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

6. Taylor expanded in z around inf 52.4%

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

   1. associate-/l* 52.0%

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

8. Simplified 52.0%

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

9. Final simplification 52.0%

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

Developer target: 93.5% 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 2023196 
(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)))