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

Percentage Accurate: 85.6% → 99.3%

Time: 10.4s

Alternatives: 12

Speedup: 1.0×

• 23.6% 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: 85.6% 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: 99.3% 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 86.2%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
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 99.0%

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

Alternative 2: 68.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z)))
        (t_2 (+ (/ x y) -2.0))
        (t_3 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_3 -5e+181)
     t_1
     (if (<= t_3 1e+144)
       t_2
       (if (<= t_3 4e+289) (/ 2.0 t) (if (<= t_3 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+181) {
		tmp = t_1;
	} else if (t_3 <= 1e+144) {
		tmp = t_2;
	} else if (t_3 <= 4e+289) {
		tmp = 2.0 / t;
	} 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 / (t * z);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+181) {
		tmp = t_1;
	} else if (t_3 <= 1e+144) {
		tmp = t_2;
	} else if (t_3 <= 4e+289) {
		tmp = 2.0 / t;
	} 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 / (t * z)
	t_2 = (x / y) + -2.0
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if t_3 <= -5e+181:
		tmp = t_1
	elif t_3 <= 1e+144:
		tmp = t_2
	elif t_3 <= 4e+289:
		tmp = 2.0 / t
	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(t * z))
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_3 <= -5e+181)
		tmp = t_1;
	elseif (t_3 <= 1e+144)
		tmp = t_2;
	elseif (t_3 <= 4e+289)
		tmp = Float64(2.0 / t);
	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 / (t * z);
	t_2 = (x / y) + -2.0;
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if (t_3 <= -5e+181)
		tmp = t_1;
	elseif (t_3 <= 1e+144)
		tmp = t_2;
	elseif (t_3 <= 4e+289)
		tmp = 2.0 / t;
	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[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+181], t$95$1, If[LessEqual[t$95$3, 1e+144], t$95$2, If[LessEqual[t$95$3, 4e+289], N[(2.0 / t), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]]

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)) < -5.0000000000000003e181 or 4.0000000000000002e289 < (/.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 96.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 96.6%

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

   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 75.9

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

   8. Applied rewrites 75.9%

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

   if -5.0000000000000003e181 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.00000000000000002e144 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 79.4%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 75.7%

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

   if 1.00000000000000002e144 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.0000000000000002e289

   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 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 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 26.1%

      \[\leadsto \frac{\frac{2}{z}}{t} \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 58.1%

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

Alternative 3: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.55 \cdot 10^{+54} \lor \neg \left(\frac{x}{y} \leq 2.4 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1.55e+54) (not (<= (/ x y) 2.4e+28)))
   (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) <= -1.55e+54) || !((x / y) <= 2.4e+28)) {
		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) <= -1.55e+54) || !(Float64(x / y) <= 2.4e+28))
		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], -1.55e+54], N[Not[LessEqual[N[(x / y), $MachinePrecision], 2.4e+28]], $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) < -1.55e54 or 2.39999999999999981e28 < (/.f64 x y)

   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 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 82.3

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

   5. Applied rewrites 82.3%

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

   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 82.3%

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

   if -1.55e54 < (/.f64 x y) < 2.39999999999999981e28

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add-rev N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

      10. count-2-rev N/A

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

   8. Applied rewrites 97.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.55 \cdot 10^{+54} \lor \neg \left(\frac{x}{y} \leq 2.4 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left({t}^{-1}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+29} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+101} \lor \neg \left(t\_1 \leq \infty\right)\right):\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

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 -5e+29) (not (or (<= t_1 5e+101) (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 <= -5e+29) || !((t_1 <= 5e+101) || !(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 <= -5e+29) || !((t_1 <= 5e+101) || !(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 <= -5e+29) or not ((t_1 <= 5e+101) 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 <= -5e+29) || !((t_1 <= 5e+101) || !(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 <= -5e+29) || ~(((t_1 <= 5e+101) || ~((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, -5e+29], N[Not[Or[LessEqual[t$95$1, 5e+101], 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)) < -5.0000000000000001e29 or 4.99999999999999989e101 < (/.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 98.4%

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

   3. Taylor expanded in 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 78.3

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

   5. Applied rewrites 78.3%

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

   if -5.0000000000000001e29 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.99999999999999989e101 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 72.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 88.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -5 \cdot 10^{+29} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 5 \cdot 10^{+101} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty\right)\right):\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{\frac{\mathsf{fma}\left(z, 2, 2\right)}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e+57)
   (+ (/ x y) (/ 2.0 (* t z)))
   (if (<= (/ x y) 5e-10)
     (/ (fma (- 1.0 t) 2.0 (/ 2.0 z)) t)
     (+ (/ x y) (/ (/ (fma z 2.0 2.0) t) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e+57) {
		tmp = (x / y) + (2.0 / (t * z));
	} else if ((x / y) <= 5e-10) {
		tmp = fma((1.0 - t), 2.0, (2.0 / z)) / t;
	} else {
		tmp = (x / y) + ((fma(z, 2.0, 2.0) / t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1e+57)
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	elseif (Float64(x / y) <= 5e-10)
		tmp = Float64(fma(Float64(1.0 - t), 2.0, Float64(2.0 / z)) / t);
	else
		tmp = Float64(Float64(x / y) + Float64(Float64(fma(z, 2.0, 2.0) / t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -1e+57], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e-10], N[(N[(N[(1.0 - t), $MachinePrecision] * 2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(N[(N[(z * 2.0 + 2.0), $MachinePrecision] / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.00000000000000005e57

   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 z around 0

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

   5. Applied rewrites 95.6%

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

   if -1.00000000000000005e57 < (/.f64 x y) < 5.00000000000000031e-10

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 5.00000000000000031e-10 < (/.f64 x y)

   1. Initial program 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 89.6%

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

Alternative 6: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+57} \lor \neg \left(\frac{x}{y} \leq 4 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1e+57) (not (<= (/ x y) 4e+26)))
   (+ (/ x y) (/ 2.0 (* t z)))
   (/ (fma (- 1.0 t) 2.0 (/ 2.0 z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -1e+57) || !((x / y) <= 4e+26)) {
		tmp = (x / y) + (2.0 / (t * z));
	} else {
		tmp = fma((1.0 - t), 2.0, (2.0 / z)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -1e+57) || !(Float64(x / y) <= 4e+26))
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	else
		tmp = Float64(fma(Float64(1.0 - t), 2.0, Float64(2.0 / z)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -1e+57], N[Not[LessEqual[N[(x / y), $MachinePrecision], 4e+26]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] * 2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.00000000000000005e57 or 4.00000000000000019e26 < (/.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 90.4%

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

   if -1.00000000000000005e57 < (/.f64 x y) < 4.00000000000000019e26

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+57} \lor \neg \left(\frac{x}{y} \leq 4 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.4 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -3.4e+55) (not (<= (/ x y) 2.2e+27)))
   (+ (/ x y) (/ 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) <= -3.4e+55) || !((x / y) <= 2.2e+27)) {
		tmp = (x / y) + (2.0 / (t * z));
	} 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) <= (-3.4d+55)) .or. (.not. ((x / y) <= 2.2d+27))) then
        tmp = (x / y) + (2.0d0 / (t * z))
    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) <= -3.4e+55) || !((x / y) <= 2.2e+27)) {
		tmp = (x / y) + (2.0 / (t * z));
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -3.4e+55) or not ((x / y) <= 2.2e+27):
		tmp = (x / y) + (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) <= -3.4e+55) || !(Float64(x / y) <= 2.2e+27))
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	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) <= -3.4e+55) || ~(((x / y) <= 2.2e+27)))
		tmp = (x / y) + (2.0 / (t * z));
	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], -3.4e+55], N[Not[LessEqual[N[(x / y), $MachinePrecision], 2.2e+27]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + N[(2.0 / 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) < -3.3999999999999998e55 or 2.1999999999999999e27 < (/.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 90.4%

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

   if -3.3999999999999998e55 < (/.f64 x y) < 2.1999999999999999e27

   1. Initial program 86.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 99.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add-rev N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

      10. count-2-rev N/A

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

   8. Applied rewrites 97.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.4 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.42 \cdot 10^{+89} \lor \neg \left(\frac{x}{y} \leq 1.05 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1.42e+89) (not (<= (/ x y) 1.05e+30)))
   (/ 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) <= -1.42e+89) || !((x / y) <= 1.05e+30)) {
		tmp = x / y;
	} 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) <= (-1.42d+89)) .or. (.not. ((x / y) <= 1.05d+30))) then
        tmp = x / y
    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) <= -1.42e+89) || !((x / y) <= 1.05e+30)) {
		tmp = x / y;
	} else {
		tmp = -2.0 - (((-2.0 / z) - 2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -1.42e+89) or not ((x / y) <= 1.05e+30):
		tmp = 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) <= -1.42e+89) || !(Float64(x / y) <= 1.05e+30))
		tmp = Float64(x / y);
	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) <= -1.42e+89) || ~(((x / y) <= 1.05e+30)))
		tmp = x / y;
	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], -1.42e+89], N[Not[LessEqual[N[(x / y), $MachinePrecision], 1.05e+30]], $MachinePrecision]], N[(x / y), $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) < -1.42e89 or 1.05e30 < (/.f64 x y)

   1. Initial program 87.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   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 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.3%

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

   if -1.42e89 < (/.f64 x y) < 1.05e30

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0

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

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add-rev N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

      10. count-2-rev N/A

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

   8. Applied rewrites 95.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.42 \cdot 10^{+89} \lor \neg \left(\frac{x}{y} \leq 1.05 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{z} - 2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-16} \lor \neg \left(z \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y} - 2, t, \frac{2}{z} - -2\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.8e-16) (not (<= z 4.2e-5)))
   (fma (/ (- 1.0 t) t) 2.0 (/ x y))
   (/ (fma (- (/ x y) 2.0) t (- (/ 2.0 z) -2.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.8e-16) || !(z <= 4.2e-5)) {
		tmp = fma(((1.0 - t) / t), 2.0, (x / y));
	} else {
		tmp = fma(((x / y) - 2.0), t, ((2.0 / z) - -2.0)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.8e-16) || !(z <= 4.2e-5))
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, Float64(x / y));
	else
		tmp = Float64(fma(Float64(Float64(x / y) - 2.0), t, Float64(Float64(2.0 / z) - -2.0)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.8e-16], N[Not[LessEqual[z, 4.2e-5]], $MachinePrecision]], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision] * t + N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000001e-16 or 4.19999999999999977e-5 < z

   1. Initial program 74.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -4.8000000000000001e-16 < z < 4.19999999999999977e-5

   1. Initial program 98.3%

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

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

   5. Applied rewrites 94.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-16} \lor \neg \left(z \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y} - 2, t, \frac{2}{z} - -2\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.4 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -3.4e+55) (not (<= (/ x y) 2.2e+27)))
   (/ x y)
   (- (/ 2.0 t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -3.4e+55) || !((x / y) <= 2.2e+27)) {
		tmp = x / y;
	} 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) <= (-3.4d+55)) .or. (.not. ((x / y) <= 2.2d+27))) then
        tmp = x / y
    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) <= -3.4e+55) || !((x / y) <= 2.2e+27)) {
		tmp = x / y;
	} else {
		tmp = (2.0 / t) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -3.4e+55) or not ((x / y) <= 2.2e+27):
		tmp = x / y
	else:
		tmp = (2.0 / t) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -3.4e+55) || !(Float64(x / y) <= 2.2e+27))
		tmp = Float64(x / y);
	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) <= -3.4e+55) || ~(((x / y) <= 2.2e+27)))
		tmp = x / y;
	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], -3.4e+55], N[Not[LessEqual[N[(x / y), $MachinePrecision], 2.2e+27]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -3.3999999999999998e55 or 2.1999999999999999e27 < (/.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.1%

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

   if -3.3999999999999998e55 < (/.f64 x y) < 2.1999999999999999e27

   1. Initial program 86.0%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.9%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.4 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.2 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -3.2e+55) (not (<= (/ x y) 2.2e+27))) (/ x y) (/ 2.0 t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -3.2e+55) || !((x / y) <= 2.2e+27)) {
		tmp = x / y;
	} 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 (((x / y) <= (-3.2d+55)) .or. (.not. ((x / y) <= 2.2d+27))) then
        tmp = x / y
    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 (((x / y) <= -3.2e+55) || !((x / y) <= 2.2e+27)) {
		tmp = x / y;
	} else {
		tmp = 2.0 / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -3.2e+55) or not ((x / y) <= 2.2e+27):
		tmp = x / y
	else:
		tmp = 2.0 / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -3.2e+55) || !(Float64(x / y) <= 2.2e+27))
		tmp = Float64(x / y);
	else
		tmp = Float64(2.0 / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -3.2e+55) || ~(((x / y) <= 2.2e+27)))
		tmp = x / y;
	else
		tmp = 2.0 / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -3.2e+55], N[Not[LessEqual[N[(x / y), $MachinePrecision], 2.2e+27]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(2.0 / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -3.2000000000000003e55 or 2.1999999999999999e27 < (/.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.1%

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

   if -3.2000000000000003e55 < (/.f64 x y) < 2.1999999999999999e27

   1. Initial program 86.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 40.7%

      \[\leadsto \frac{\frac{2}{z}}{t} \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 26.2%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.2 \cdot 10^{+55} \lor \neg \left(\frac{x}{y} \leq 2.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 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 = x / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
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 70.9

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

5. Applied rewrites 70.9%

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

7. Applied rewrites 67.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 38.7%

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

11. Final simplification 38.7%

    \[\leadsto \frac{x}{y} \]
12. Add Preprocessing

Developer Target 1: 99.0% 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 2024326 
(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))))