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

Percentage Accurate: 86.4% → 98.0%

Time: 10.7s

Alternatives: 14

Speedup: 1.0×

• 23.9% 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: 86.4% 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: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.2 \cdot 10^{+17} \lor \neg \left(\frac{x}{y} \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -6.2e+17) (not (<= (/ x y) 1.1e-15)))
   (+ (/ x y) (/ (fma 2.0 z 2.0) (* t z)))
   (- -2.0 (/ (- (/ -2.0 z) 2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -6.2e+17) || !((x / y) <= 1.1e-15)) {
		tmp = (x / y) + (fma(2.0, z, 2.0) / (t * z));
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -6.2e+17) || !(Float64(x / y) <= 1.1e-15))
		tmp = Float64(Float64(x / y) + Float64(fma(2.0, z, 2.0) / Float64(t * z)));
	else
		tmp = Float64(-2.0 - Float64(Float64(Float64(-2.0 / z) - 2.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -6.2e+17], N[Not[LessEqual[N[(x / y), $MachinePrecision], 1.1e-15]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 - N[(N[(N[(-2.0 / z), $MachinePrecision] - 2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -6.2e17 or 1.09999999999999993e-15 < (/.f64 x y)

   1. Initial program 90.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -6.2e17 < (/.f64 x y) < 1.09999999999999993e-15

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.2 \cdot 10^{+17} \lor \neg \left(\frac{x}{y} \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.1 \cdot 10^{+39} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1.1e+39) (not (<= (/ x y) 6.6e+14)))
   (fma (pow t -1.0) 2.0 (/ x y))
   (- -2.0 (/ (- (/ -2.0 z) 2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -1.1e+39) || !((x / y) <= 6.6e+14)) {
		tmp = fma(pow(t, -1.0), 2.0, (x / y));
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -1.1e+39) || !(Float64(x / y) <= 6.6e+14))
		tmp = fma((t ^ -1.0), 2.0, Float64(x / y));
	else
		tmp = Float64(-2.0 - Float64(Float64(Float64(-2.0 / z) - 2.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -1.1e+39], N[Not[LessEqual[N[(x / y), $MachinePrecision], 6.6e+14]], $MachinePrecision]], N[(N[Power[t, -1.0], $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 - N[(N[(N[(-2.0 / z), $MachinePrecision] - 2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.1000000000000001e39 or 6.6e14 < (/.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{t}, 2, \frac{x}{y}\right) \]

   if -1.1000000000000001e39 < (/.f64 x y) < 6.6e14

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.1 \cdot 10^{+39} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{y} \leq 6.6 \cdot 10^{+14}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6.8e+41)
   (+ (/ x y) (/ 2.0 (* t z)))
   (if (<= (/ x y) 6.6e+14)
     (- -2.0 (/ (- (/ -2.0 z) 2.0) t))
     (fma (pow t -1.0) 2.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6.8e+41) {
		tmp = (x / y) + (2.0 / (t * z));
	} else if ((x / y) <= 6.6e+14) {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	} else {
		tmp = fma(pow(t, -1.0), 2.0, (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6.8e+41)
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	elseif (Float64(x / y) <= 6.6e+14)
		tmp = Float64(-2.0 - Float64(Float64(Float64(-2.0 / z) - 2.0) / t));
	else
		tmp = fma((t ^ -1.0), 2.0, Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -6.8e+41], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 6.6e+14], N[(-2.0 - N[(N[(N[(-2.0 / z), $MachinePrecision] - 2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[Power[t, -1.0], $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -6.79999999999999996e41

   1. Initial program 93.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 86.1%

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

   if -6.79999999999999996e41 < (/.f64 x y) < 6.6e14

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if 6.6e14 < (/.f64 x y)

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{y} \leq 6.6 \cdot 10^{+14}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+37} \lor \neg \left(t\_1 \leq \infty\right)\right):\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (or (<= t_1 -2e+53) (not (or (<= t_1 2e+37) (not (<= t_1 INFINITY)))))
     (/ (- (/ 2.0 z) -2.0) t)
     (+ (/ x y) -2.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+37) || !(t_1 <= ((double) INFINITY)))) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+37) || !(t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if (t_1 <= -2e+53) or not ((t_1 <= 2e+37) or not (t_1 <= math.inf)):
		tmp = ((2.0 / z) - -2.0) / t
	else:
		tmp = (x / y) + -2.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+37) || !(t_1 <= Inf)))
		tmp = Float64(Float64(Float64(2.0 / z) - -2.0) / t);
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if ((t_1 <= -2e+53) || ~(((t_1 <= 2e+37) || ~((t_1 <= Inf)))))
		tmp = ((2.0 / z) - -2.0) / t;
	else
		tmp = (x / y) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+53], N[Not[Or[LessEqual[t$95$1, 2e+37], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $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)) < -2e53 or 1.99999999999999991e37 < (/.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 99.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. +-commutative N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 N/A

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

      21. associate-*r/ N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -2e53 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.99999999999999991e37 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 76.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 93.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -2 \cdot 10^{+53} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 2 \cdot 10^{+37} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty\right)\right):\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+83} \lor \neg \left(t\_1 \leq \infty\right)\right):\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (or (<= t_1 -2e+53) (not (or (<= t_1 2e+83) (not (<= t_1 INFINITY)))))
     (/ 2.0 (* t z))
     (+ (/ x y) -2.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+83) || !(t_1 <= ((double) INFINITY)))) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+83) || !(t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if (t_1 <= -2e+53) or not ((t_1 <= 2e+83) or not (t_1 <= math.inf)):
		tmp = 2.0 / (t * z)
	else:
		tmp = (x / y) + -2.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if ((t_1 <= -2e+53) || !((t_1 <= 2e+83) || !(t_1 <= Inf)))
		tmp = Float64(2.0 / Float64(t * z));
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if ((t_1 <= -2e+53) || ~(((t_1 <= 2e+83) || ~((t_1 <= Inf)))))
		tmp = 2.0 / (t * z);
	else
		tmp = (x / y) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+53], N[Not[Or[LessEqual[t$95$1, 2e+83], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]]], $MachinePrecision]], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $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)) < -2e53 or 2.00000000000000006e83 < (/.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 99.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.1

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

   8. Applied rewrites 54.1%

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

   if -2e53 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2.00000000000000006e83 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 78.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 88.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -2 \cdot 10^{+53} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 2 \cdot 10^{+83} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty\right)\right):\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x y) (/.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 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if +inf.0 < (+.f64 (/.f64 x y) (/.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 96.3%

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

Alternative 7: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{y} \leq 6.6 \cdot 10^{+14}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6.8e+41)
   (+ (/ x y) (/ 2.0 (* t z)))
   (if (<= (/ x y) 6.6e+14)
     (- -2.0 (/ (- (/ -2.0 z) 2.0) t))
     (fma (/ (- 1.0 t) t) 2.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6.8e+41) {
		tmp = (x / y) + (2.0 / (t * z));
	} else if ((x / y) <= 6.6e+14) {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	} else {
		tmp = fma(((1.0 - t) / t), 2.0, (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6.8e+41)
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	elseif (Float64(x / y) <= 6.6e+14)
		tmp = Float64(-2.0 - Float64(Float64(Float64(-2.0 / z) - 2.0) / t));
	else
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -6.8e+41], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 6.6e+14], N[(-2.0 - N[(N[(N[(-2.0 / z), $MachinePrecision] - 2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -6.79999999999999996e41

   1. Initial program 93.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 86.1%

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

   if -6.79999999999999996e41 < (/.f64 x y) < 6.6e14

   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 x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if 6.6e14 < (/.f64 x y)

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

Alternative 8: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.5 \cdot 10^{+156} \lor \neg \left(\frac{x}{y} \leq 1.15 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -3.5e+156) (not (<= (/ x y) 1.15e+64)))
   (+ (/ x y) -2.0)
   (- -2.0 (/ (- (/ -2.0 z) 2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -3.5e+156) || !((x / y) <= 1.15e+64)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	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) <= (-3.5d+156)) .or. (.not. ((x / y) <= 1.15d+64))) then
        tmp = (x / y) + (-2.0d0)
    else
        tmp = (-2.0d0) - ((((-2.0d0) / z) - 2.0d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -3.5e+156) || !((x / y) <= 1.15e+64)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -3.5e+156) or not ((x / y) <= 1.15e+64):
		tmp = (x / y) + -2.0
	else:
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -3.5e+156) || !(Float64(x / y) <= 1.15e+64))
		tmp = Float64(Float64(x / y) + -2.0);
	else
		tmp = Float64(-2.0 - Float64(Float64(Float64(-2.0 / z) - 2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -3.5e+156) || ~(((x / y) <= 1.15e+64)))
		tmp = (x / y) + -2.0;
	else
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -3.5e+156], N[Not[LessEqual[N[(x / y), $MachinePrecision], 1.15e+64]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision], N[(-2.0 - N[(N[(N[(-2.0 / z), $MachinePrecision] - 2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -3.5000000000000003e156 or 1.15e64 < (/.f64 x y)

   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 t around inf

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

   5. Applied rewrites 79.4%

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

   if -3.5000000000000003e156 < (/.f64 x y) < 1.15e64

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.5 \cdot 10^{+156} \lor \neg \left(\frac{x}{y} \leq 1.15 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.1 \cdot 10^{+39} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{t} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1.1e+39) (not (<= (/ x y) 6.6e+14)))
   (+ (/ x y) -2.0)
   (- (/ (/ 2.0 z) t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -1.1e+39) || !((x / y) <= 6.6e+14)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = ((2.0 / z) / t) - 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 (((x / y) <= (-1.1d+39)) .or. (.not. ((x / y) <= 6.6d+14))) then
        tmp = (x / y) + (-2.0d0)
    else
        tmp = ((2.0d0 / z) / t) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -1.1e+39) || !((x / y) <= 6.6e+14)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = ((2.0 / z) / t) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -1.1e+39) or not ((x / y) <= 6.6e+14):
		tmp = (x / y) + -2.0
	else:
		tmp = ((2.0 / z) / t) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -1.1e+39) || !(Float64(x / y) <= 6.6e+14))
		tmp = Float64(Float64(x / y) + -2.0);
	else
		tmp = Float64(Float64(Float64(2.0 / z) / t) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -1.1e+39) || ~(((x / y) <= 6.6e+14)))
		tmp = (x / y) + -2.0;
	else
		tmp = ((2.0 / z) / t) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -1.1e+39], N[Not[LessEqual[N[(x / y), $MachinePrecision], 6.6e+14]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(N[(2.0 / z), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.1000000000000001e39 or 6.6e14 < (/.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 t around inf

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

   5. Applied rewrites 70.8%

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

   if -1.1000000000000001e39 < (/.f64 x y) < 6.6e14

   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 z around 0

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.6

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

   8. Applied rewrites 64.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. fp-cancel-sign-sub-inv N/A

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

       7. +-commutative N/A

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

       8. div-add N/A

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

       9. associate-/l* N/A

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

       10. *-inverses N/A

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

       11. metadata-eval N/A

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

       12. distribute-lft-in N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. fp-cancel-sub-sign-inv N/A

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

       16. associate-+l- N/A

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

       17. associate-*r/ N/A

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

       18. metadata-eval N/A

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in z around 0

       \[\leadsto \frac{\frac{2}{z}}{t} - 2 \]
   13. Step-by-step derivation

   14. Applied rewrites 77.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.1 \cdot 10^{+39} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -92000000000 \lor \neg \left(\frac{x}{y} \leq 23000000000\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -92000000000.0) (not (<= (/ x y) 23000000000.0)))
   (+ (/ x y) -2.0)
   (- (/ 2.0 t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -92000000000.0) || !((x / y) <= 23000000000.0)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = (2.0 / t) - 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 (((x / y) <= (-92000000000.0d0)) .or. (.not. ((x / y) <= 23000000000.0d0))) then
        tmp = (x / y) + (-2.0d0)
    else
        tmp = (2.0d0 / t) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -92000000000.0) || !((x / y) <= 23000000000.0)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = (2.0 / t) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -92000000000.0) or not ((x / y) <= 23000000000.0):
		tmp = (x / y) + -2.0
	else:
		tmp = (2.0 / t) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -92000000000.0) || !(Float64(x / y) <= 23000000000.0))
		tmp = Float64(Float64(x / y) + -2.0);
	else
		tmp = Float64(Float64(2.0 / t) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -92000000000.0) || ~(((x / y) <= 23000000000.0)))
		tmp = (x / y) + -2.0;
	else
		tmp = (2.0 / t) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -92000000000.0], N[Not[LessEqual[N[(x / y), $MachinePrecision], 23000000000.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -9.2e10 or 2.3e10 < (/.f64 x y)

   1. Initial program 89.8%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 69.0%

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

   if -9.2e10 < (/.f64 x y) < 2.3e10

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -92000000000 \lor \neg \left(\frac{x}{y} \leq 23000000000\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. associate-*l/ N/A

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

   13. distribute-rgt-out N/A

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

   14. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

Alternative 12: 35.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -58 \lor \neg \left(t \leq 1\right):\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -58.0) (not (<= t 1.0))) -2.0 (/ 2.0 t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -58.0) || !(t <= 1.0)) {
		tmp = -2.0;
	} else {
		tmp = 2.0 / t;
	}
	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 <= (-58.0d0)) .or. (.not. (t <= 1.0d0))) then
        tmp = -2.0d0
    else
        tmp = 2.0d0 / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -58.0) || !(t <= 1.0)) {
		tmp = -2.0;
	} else {
		tmp = 2.0 / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -58.0) or not (t <= 1.0):
		tmp = -2.0
	else:
		tmp = 2.0 / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -58.0) || !(t <= 1.0))
		tmp = -2.0;
	else
		tmp = Float64(2.0 / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -58.0) || ~((t <= 1.0)))
		tmp = -2.0;
	else
		tmp = 2.0 / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -58.0], N[Not[LessEqual[t, 1.0]], $MachinePrecision]], -2.0, N[(2.0 / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -58 or 1 < t

   1. Initial program 79.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 61.4

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. fp-cancel-sign-sub-inv N/A

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

       7. +-commutative N/A

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

       8. div-add N/A

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

       9. associate-/l* N/A

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

       10. *-inverses N/A

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

       11. metadata-eval N/A

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

       12. distribute-lft-in N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. fp-cancel-sub-sign-inv N/A

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

       16. associate-+l- N/A

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

       17. associate-*r/ N/A

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

       18. metadata-eval N/A

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

   11. Applied rewrites 56.0%

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

   12. Taylor expanded in t around inf

       \[\leadsto -2 \]
   13. Step-by-step derivation

   14. Applied rewrites 39.5%

       \[\leadsto -2 \]

   if -58 < t < 1

   1. Initial program 99.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. +-commutative N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 N/A

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

      21. associate-*r/ N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 36.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -58 \lor \neg \left(t \leq 1\right):\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (2.0 / t) - 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 / t) - 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 69.8

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

5. Applied rewrites 69.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 38.9%

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

Alternative 14: 20.1% 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 89.3%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 62.3%

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

6. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 80.1

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

8. Applied rewrites 80.1%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. fp-cancel-sign-sub-inv N/A

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

    3. metadata-eval N/A

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

    4. *-lft-identity N/A

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

    5. metadata-eval N/A

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

    6. fp-cancel-sign-sub-inv N/A

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

    7. +-commutative N/A

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

    8. div-add N/A

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

    9. associate-/l* N/A

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

    10. *-inverses N/A

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

    11. metadata-eval N/A

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

    12. distribute-lft-in N/A

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

    13. metadata-eval N/A

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

    14. metadata-eval N/A

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

    15. fp-cancel-sub-sign-inv N/A

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

    16. associate-+l- N/A

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

    17. associate-*r/ N/A

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

    18. metadata-eval N/A

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

11. Applied rewrites 68.9%

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

12. Taylor expanded in t around inf

    \[\leadsto -2 \]
13. Step-by-step derivation

14. Applied rewrites 21.0%

    \[\leadsto -2 \]
15. 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 2024338 
(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))))