Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Percentage Accurate: 86.2% → 98.3%

Time: 13.1s

Alternatives: 17

Speedup: 0.7×

• 23.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;t\_1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_1 2e+285)
     (+ t_1 (/ x y))
     (/ (fma y (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) x) y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_1 <= 2e+285) {
		tmp = t_1 + (x / y);
	} else {
		tmp = fma(y, fma((2.0 / (z * t)), (z + 1.0), -2.0), x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_1 <= 2e+285)
		tmp = Float64(t_1 + Float64(x / y));
	else
		tmp = Float64(fma(y, fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0), x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+285], N[(t$95$1 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2e285

   1. Initial program 99.8%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   if 2e285 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 32.6%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right), x\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1.999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq 10^{+69}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (/ 2.0 z)) t))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -2e+50)
     t_1
     (if (<= t_2 -1.999999)
       (/ (fma y -2.0 x) y)
       (if (<= t_2 1e+69)
         (+ (/ x y) (/ 2.0 t))
         (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -2e+50) {
		tmp = t_1;
	} else if (t_2 <= -1.999999) {
		tmp = fma(y, -2.0, x) / y;
	} else if (t_2 <= 1e+69) {
		tmp = (x / y) + (2.0 / t);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(2.0 / z)) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+50)
		tmp = t_1;
	elseif (t_2 <= -1.999999)
		tmp = Float64(fma(y, -2.0, x) / y);
	elseif (t_2 <= 1e+69)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+50], t$95$1, If[LessEqual[t$95$2, -1.999999], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, 1e+69], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2.0000000000000002e50 or 1.0000000000000001e69 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{z} + t \cdot \left(\frac{x}{y} - 2\right)\right)}{t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{2 + 2 \cdot \frac{1}{z}}}{t} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + 2 \cdot \frac{1}{z}}}{t} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{z}}}{t} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{\color{blue}{2}}{z}}{t} \]

      4. lower-/.f64 79.1

         \[\leadsto \frac{2 + \color{blue}{\frac{2}{z}}}{t} \]

   8. Applied rewrites 79.1%

      \[\leadsto \frac{\color{blue}{2 + \frac{2}{z}}}{t} \]

   if -2.0000000000000002e50 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.99999900000000008

   1. Initial program 99.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{-2 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y - -1 \cdot x}}{y} \]

      5. *-lft-identity N/A

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

      6. *-inverses N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot y - \color{blue}{y \cdot \frac{-1 \cdot x}{y}}}{y} \]

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot y + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}}{y} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot -2} + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right)}{y} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{x}{y}\right)\right)}\right)\right)}{y} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y \cdot -2 + \color{blue}{y \cdot \frac{x}{y}}}{y} \]

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}{y} \]

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 92.5%

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

   if -1.99999900000000008 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.0000000000000001e69

   1. Initial program 99.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 83.2

         \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]

   8. Applied rewrites 83.2%

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]

   if +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 0.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -1.999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 10^{+69}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1.999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq 10^{+69}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -2e+50)
     t_1
     (if (<= t_2 -1.999999)
       (/ (fma y -2.0 x) y)
       (if (<= t_2 1e+69)
         (+ (/ x y) (/ 2.0 t))
         (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, 2.0) / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -2e+50) {
		tmp = t_1;
	} else if (t_2 <= -1.999999) {
		tmp = fma(y, -2.0, x) / y;
	} else if (t_2 <= 1e+69) {
		tmp = (x / y) + (2.0 / t);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, 2.0) / Float64(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+50)
		tmp = t_1;
	elseif (t_2 <= -1.999999)
		tmp = Float64(fma(y, -2.0, x) / y);
	elseif (t_2 <= 1e+69)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+50], t$95$1, If[LessEqual[t$95$2, -1.999999], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, 1e+69], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2.0000000000000002e50 or 1.0000000000000001e69 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]

   5. Applied rewrites 79.0%

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

   if -2.0000000000000002e50 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.99999900000000008

   1. Initial program 99.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{-2 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y - -1 \cdot x}}{y} \]

      5. *-lft-identity N/A

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

      6. *-inverses N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot y - \color{blue}{y \cdot \frac{-1 \cdot x}{y}}}{y} \]

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot y + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}}{y} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot -2} + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right)}{y} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{x}{y}\right)\right)}\right)\right)}{y} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y \cdot -2 + \color{blue}{y \cdot \frac{x}{y}}}{y} \]

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}{y} \]

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 92.5%

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

   if -1.99999900000000008 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.0000000000000001e69

   1. Initial program 99.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 83.2

         \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]

   8. Applied rewrites 83.2%

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]

   if +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 0.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -1.999999:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 10^{+69}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -2e+50)
     t_1
     (if (<= t_2 2e+22)
       (/ (fma y -2.0 x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, 2.0) / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -2e+50) {
		tmp = t_1;
	} else if (t_2 <= 2e+22) {
		tmp = fma(y, -2.0, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, 2.0) / Float64(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+50)
		tmp = t_1;
	elseif (t_2 <= 2e+22)
		tmp = Float64(fma(y, -2.0, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+50], t$95$1, If[LessEqual[t$95$2, 2e+22], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2.0000000000000002e50 or 2e22 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]

   5. Applied rewrites 76.8%

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

   if -2.0000000000000002e50 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2e22

   1. Initial program 99.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{-2 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y - -1 \cdot x}}{y} \]

      5. *-lft-identity N/A

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

      6. *-inverses N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot y - \color{blue}{y \cdot \frac{-1 \cdot x}{y}}}{y} \]

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot y + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}}{y} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot -2} + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right)}{y} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{x}{y}\right)\right)}\right)\right)}{y} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y \cdot -2 + \color{blue}{y \cdot \frac{x}{y}}}{y} \]

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}{y} \]

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 91.3%

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

   if +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 0.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -5e+59)
     t_1
     (if (<= t_2 1e+69)
       (/ (fma y -2.0 x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -5e+59) {
		tmp = t_1;
	} else if (t_2 <= 1e+69) {
		tmp = fma(y, -2.0, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+59)
		tmp = t_1;
	elseif (t_2 <= 1e+69)
		tmp = Float64(fma(y, -2.0, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+59], t$95$1, If[LessEqual[t$95$2, 1e+69], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999997e59 or 1.0000000000000001e69 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

      2. lower-*.f64 56.3

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

   if -4.9999999999999997e59 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.0000000000000001e69

   1. Initial program 99.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{-2 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y - -1 \cdot x}}{y} \]

      5. *-lft-identity N/A

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

      6. *-inverses N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot y - \color{blue}{y \cdot \frac{-1 \cdot x}{y}}}{y} \]

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot y + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}}{y} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot -2} + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right)}{y} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{x}{y}\right)\right)}\right)\right)}{y} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y \cdot -2 + \color{blue}{y \cdot \frac{x}{y}}}{y} \]

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}{y} \]

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 83.5%

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

   if +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 0.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t)))
        (t_2 (+ (/ x y) -2.0))
        (t_3 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_3 -5e+59)
     t_1
     (if (<= t_3 1e+69) t_2 (if (<= t_3 INFINITY) t_1 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_3 <= -5e+59) {
		tmp = t_1;
	} else if (t_3 <= 1e+69) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_3 <= -5e+59) {
		tmp = t_1;
	} else if (t_3 <= 1e+69) {
		tmp = t_2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (z * t)
	t_2 = (x / y) + -2.0
	t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t)
	tmp = 0
	if t_3 <= -5e+59:
		tmp = t_1
	elif t_3 <= 1e+69:
		tmp = t_2
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_3 <= -5e+59)
		tmp = t_1;
	elseif (t_3 <= 1e+69)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (z * t);
	t_2 = (x / y) + -2.0;
	t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	tmp = 0.0;
	if (t_3 <= -5e+59)
		tmp = t_1;
	elseif (t_3 <= 1e+69)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+59], t$95$1, If[LessEqual[t$95$3, 1e+69], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999997e59 or 1.0000000000000001e69 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

      2. lower-*.f64 56.3

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

   if -4.9999999999999997e59 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.0000000000000001e69 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 79.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 10^{+69}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + t\_1\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z + 1, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t))) (t_2 (+ (/ x y) t_1)))
   (if (<= (/ x y) -5e+39)
     t_2
     (if (<= (/ x y) 1e-22)
       (fma t_1 (+ z 1.0) -2.0)
       (if (<= (/ x y) 5e+63) (+ (/ x y) (+ -2.0 (/ 2.0 t))) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + t_1;
	double tmp;
	if ((x / y) <= -5e+39) {
		tmp = t_2;
	} else if ((x / y) <= 1e-22) {
		tmp = fma(t_1, (z + 1.0), -2.0);
	} else if ((x / y) <= 5e+63) {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	t_2 = Float64(Float64(x / y) + t_1)
	tmp = 0.0
	if (Float64(x / y) <= -5e+39)
		tmp = t_2;
	elseif (Float64(x / y) <= 1e-22)
		tmp = fma(t_1, Float64(z + 1.0), -2.0);
	elseif (Float64(x / y) <= 5e+63)
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+39], t$95$2, If[LessEqual[N[(x / y), $MachinePrecision], 1e-22], N[(t$95$1 * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e+63], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -5.00000000000000015e39 or 5.00000000000000011e63 < (/.f64 x y)

   1. Initial program 87.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t \cdot z}} \]

      2. lower-*.f64 95.4

         \[\leadsto \frac{x}{y} + \frac{2}{\color{blue}{t \cdot z}} \]

   5. Applied rewrites 95.4%

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t \cdot z}} \]

   if -5.00000000000000015e39 < (/.f64 x y) < 1e-22

   1. Initial program 87.1%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 96.2%

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

   if 1e-22 < (/.f64 x y) < 5.00000000000000011e63

   1. Initial program 94.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -20:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y} + -2, 2 + \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma y (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) x) y)))
   (if (<= (/ x y) -20.0)
     t_1
     (if (<= (/ x y) 4e+49)
       (/ (fma t (+ (/ x y) -2.0) (+ 2.0 (/ 2.0 z))) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, fma((2.0 / (z * t)), (z + 1.0), -2.0), x) / y;
	double tmp;
	if ((x / y) <= -20.0) {
		tmp = t_1;
	} else if ((x / y) <= 4e+49) {
		tmp = fma(t, ((x / y) + -2.0), (2.0 + (2.0 / z))) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(y, fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0), x) / y)
	tmp = 0.0
	if (Float64(x / y) <= -20.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 4e+49)
		tmp = Float64(fma(t, Float64(Float64(x / y) + -2.0), Float64(2.0 + Float64(2.0 / z))) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -20.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 4e+49], N[(N[(t * N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision] + N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -20 or 3.99999999999999979e49 < (/.f64 x y)

   1. Initial program 88.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if -20 < (/.f64 x y) < 3.99999999999999979e49

   1. Initial program 87.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{z} + t \cdot \left(\frac{x}{y} - 2\right)\right)}{t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -20:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y} + -2, 2 + \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ t_2 := \frac{\mathsf{fma}\left(y, t\_1, x\right)}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0)) (t_2 (/ (fma y t_1 x) y)))
   (if (<= (/ x y) -2e-12) t_2 (if (<= (/ x y) 5e-7) t_1 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	double t_2 = fma(y, t_1, x) / y;
	double tmp;
	if ((x / y) <= -2e-12) {
		tmp = t_2;
	} else if ((x / y) <= 5e-7) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0)
	t_2 = Float64(fma(y, t_1, x) / y)
	tmp = 0.0
	if (Float64(x / y) <= -2e-12)
		tmp = t_2;
	elseif (Float64(x / y) <= 5e-7)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t$95$1 + x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -2e-12], t$95$2, If[LessEqual[N[(x / y), $MachinePrecision], 5e-7], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.99999999999999996e-12 or 4.99999999999999977e-7 < (/.f64 x y)

   1. Initial program 88.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.6%

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

   if -1.99999999999999996e-12 < (/.f64 x y) < 4.99999999999999977e-7

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right), x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (+ -2.0 (/ 2.0 t)))))
   (if (<= (/ x y) -1e-5)
     t_1
     (if (<= (/ x y) 1e-22) (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (-2.0 + (2.0 / t));
	double tmp;
	if ((x / y) <= -1e-5) {
		tmp = t_1;
	} else if ((x / y) <= 1e-22) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)))
	tmp = 0.0
	if (Float64(x / y) <= -1e-5)
		tmp = t_1;
	elseif (Float64(x / y) <= 1e-22)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e-5], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 1e-22], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.00000000000000008e-5 or 1e-22 < (/.f64 x y)

   1. Initial program 88.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -1.00000000000000008e-5 < (/.f64 x y) < 1e-22

   1. Initial program 87.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e+226)
   (/ x y)
   (if (<= (/ x y) 1.0)
     (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0)
     (+ (/ x y) (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e+226) {
		tmp = x / y;
	} else if ((x / y) <= 1.0) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = (x / y) + (2.0 / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e+226)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 1.0)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e+226], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1.0], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.99999999999999992e226

   1. Initial program 77.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -1.99999999999999992e226 < (/.f64 x y) < 1

   1. Initial program 88.6%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 89.5%

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

   if 1 < (/.f64 x y)

   1. Initial program 90.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 82.6

         \[\leadsto \frac{x}{y} + \color{blue}{\frac{2}{t}} \]

   8. Applied rewrites 82.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{z \cdot t} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e-5)
   (+ (/ x y) -2.0)
   (if (<= (/ x y) 5e-7) (+ (/ 2.0 (* z t)) -2.0) (/ (fma y -2.0 x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e-5) {
		tmp = (x / y) + -2.0;
	} else if ((x / y) <= 5e-7) {
		tmp = (2.0 / (z * t)) + -2.0;
	} else {
		tmp = fma(y, -2.0, x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1e-5)
		tmp = Float64(Float64(x / y) + -2.0);
	elseif (Float64(x / y) <= 5e-7)
		tmp = Float64(Float64(2.0 / Float64(z * t)) + -2.0);
	else
		tmp = Float64(fma(y, -2.0, x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -1e-5], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e-7], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.00000000000000008e-5

   1. Initial program 87.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 66.7%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]

   if -1.00000000000000008e-5 < (/.f64 x y) < 4.99999999999999977e-7

   1. Initial program 87.1%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{2}{t \cdot z}, \color{blue}{1}, -2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

      \[\leadsto \mathsf{fma}\left(\frac{2}{t \cdot z}, \color{blue}{1}, -2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \cdot 1 + -2 \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \cdot 1 + -2 \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} \cdot 1 + -2} \]

      4. *-rgt-identity 75.9

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} + -2 \]

   10. Applied rewrites 75.9%

       \[\leadsto \color{blue}{\frac{2}{t \cdot z} + -2} \]

   if 4.99999999999999977e-7 < (/.f64 x y)

   1. Initial program 89.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x + -2 \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{-2 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot y - -1 \cdot x}}{y} \]

      5. *-lft-identity N/A

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

      6. *-inverses N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot y - \color{blue}{y \cdot \frac{-1 \cdot x}{y}}}{y} \]

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot y + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}}{y} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot -2} + \left(\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)\right)}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right)}{y} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{x}{y}\right)\right)}\right)\right)}{y} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y \cdot -2 + \color{blue}{y \cdot \frac{x}{y}}}{y} \]

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}{y} \]

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 76.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{z \cdot t} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.00185:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= (/ x y) -5e-15)
     t_1
     (if (<= (/ x y) 0.00185) (+ -2.0 (/ 2.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if ((x / y) <= -5e-15) {
		tmp = t_1;
	} else if ((x / y) <= 0.00185) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) + (-2.0d0)
    if ((x / y) <= (-5d-15)) then
        tmp = t_1
    else if ((x / y) <= 0.00185d0) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if ((x / y) <= -5e-15) {
		tmp = t_1;
	} else if ((x / y) <= 0.00185) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) + -2.0
	tmp = 0
	if (x / y) <= -5e-15:
		tmp = t_1
	elif (x / y) <= 0.00185:
		tmp = -2.0 + (2.0 / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (Float64(x / y) <= -5e-15)
		tmp = t_1;
	elseif (Float64(x / y) <= 0.00185)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) + -2.0;
	tmp = 0.0;
	if ((x / y) <= -5e-15)
		tmp = t_1;
	elseif ((x / y) <= 0.00185)
		tmp = -2.0 + (2.0 / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e-15], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 0.00185], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4.99999999999999999e-15 or 0.0018500000000000001 < (/.f64 x y)

   1. Initial program 87.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 71.5%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]

   if -4.99999999999999999e-15 < (/.f64 x y) < 0.0018500000000000001

   1. Initial program 87.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{t} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{t} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{t} + \color{blue}{-2} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{t} + -2} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{t}} + -2 \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{t} + -2 \]

      6. lower-/.f64 63.2

         \[\leadsto \color{blue}{\frac{2}{t}} + -2 \]

   8. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{2}{t} + -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 0.00185:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.25:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -8.5e+38)
   (/ x y)
   (if (<= (/ x y) 2.25) (+ -2.0 (/ 2.0 t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -8.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 2.25) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x / y) <= (-8.5d+38)) then
        tmp = x / y
    else if ((x / y) <= 2.25d0) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -8.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 2.25) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -8.5e+38:
		tmp = x / y
	elif (x / y) <= 2.25:
		tmp = -2.0 + (2.0 / t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -8.5e+38)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2.25)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -8.5e+38)
		tmp = x / y;
	elseif ((x / y) <= 2.25)
		tmp = -2.0 + (2.0 / t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -8.5e+38], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2.25], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -8.4999999999999997e38 or 2.25 < (/.f64 x y)

   1. Initial program 88.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -8.4999999999999997e38 < (/.f64 x y) < 2.25

   1. Initial program 86.6%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{t} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{t} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{t} + \color{blue}{-2} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{t} + -2} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{t}} + -2 \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{t} + -2 \]

      6. lower-/.f64 60.4

         \[\leadsto \color{blue}{\frac{2}{t}} + -2 \]

   8. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\frac{2}{t} + -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.25:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2.0) (/ x y) (if (<= (/ x y) 2.0) -2.0 (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2.0) {
		tmp = x / y;
	} else if ((x / y) <= 2.0) {
		tmp = -2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x / y) <= (-2.0d0)) then
        tmp = x / y
    else if ((x / y) <= 2.0d0) then
        tmp = -2.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2.0) {
		tmp = x / y;
	} else if ((x / y) <= 2.0) {
		tmp = -2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -2.0:
		tmp = x / y
	elif (x / y) <= 2.0:
		tmp = -2.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2.0)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2.0)
		tmp = -2.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -2.0)
		tmp = x / y;
	elseif ((x / y) <= 2.0)
		tmp = -2.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2.0], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2.0], -2.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2 or 2 < (/.f64 x y)

   1. Initial program 89.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -2 < (/.f64 x y) < 2

   1. Initial program 85.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-2} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.9%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1100000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.0) -2.0 (if (<= t 1100000.0) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.0) {
		tmp = -2.0;
	} else if (t <= 1100000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t <= 1100000.0d0) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.0) {
		tmp = -2.0;
	} else if (t <= 1100000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.0:
		tmp = -2.0
	elif t <= 1100000.0:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.0)
		tmp = -2.0;
	elseif (t <= 1100000.0)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.0)
		tmp = -2.0;
	elseif (t <= 1100000.0)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.0], -2.0, If[LessEqual[t, 1100000.0], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -1 or 1.1e6 < t

   1. Initial program 79.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-2} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto \color{blue}{-2} \]

   if -1 < t < 1.1e6

   1. Initial program 97.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{y} + \color{blue}{2 \cdot \frac{1 - t}{t}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \color{blue}{\left(\frac{1}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto \frac{x}{y} + 2 \cdot \left(\frac{1}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 33.5

         \[\leadsto \color{blue}{\frac{2}{t}} \]

   8. Applied rewrites 33.5%

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

Alternative 17: 20.4% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 -2.0)

C

double code(double x, double y, double z, double t) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -2.0d0
end function

Java

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

Python

def code(x, y, z, t):
	return -2.0

Julia

function code(x, y, z, t)
	return -2.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_, t_] := -2.0

Derivation

1. Initial program 87.7%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]

5. Applied rewrites 62.3%

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

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-2} \]
7. Step-by-step derivation

8. Applied rewrites 19.7%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Developer Target 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((2.0d0 / z) + 2.0d0) / t) - (2.0d0 - (x / y))
end function

Java

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

Python

def code(x, y, z, t):
	return (((2.0 / z) + 2.0) / t) - (2.0 - (x / y))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(2.0 / z) + 2.0) / t) - Float64(2.0 - Float64(x / y)))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y))))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))