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

Percentage Accurate: 86.3% → 99.1%

Time: 13.6s

Alternatives: 16

Speedup: 1.2×

• 23.2% 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: 86.3% 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: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Taylor expanded in z around 0

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

5. Simplified 99.0%

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

Alternative 2: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} + \frac{2}{t}\\ t_4 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -10000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 6 \cdot 10^{+209}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (/ 2.0 z)) t))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* t z)))
        (t_3 (+ (/ x y) (/ 2.0 t)))
        (t_4 (+ (/ x y) -2.0)))
   (if (<= t_2 -2e+168)
     t_1
     (if (<= t_2 -10000000000000.0)
       t_3
       (if (<= t_2 -1.0)
         t_4
         (if (<= t_2 6e+209) t_3 (if (<= t_2 INFINITY) t_1 t_4)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) + (2.0 / t);
	double t_4 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -2e+168) {
		tmp = t_1;
	} else if (t_2 <= -10000000000000.0) {
		tmp = t_3;
	} else if (t_2 <= -1.0) {
		tmp = t_4;
	} else if (t_2 <= 6e+209) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) + (2.0 / t);
	double t_4 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -2e+168) {
		tmp = t_1;
	} else if (t_2 <= -10000000000000.0) {
		tmp = t_3;
	} else if (t_2 <= -1.0) {
		tmp = t_4;
	} else if (t_2 <= 6e+209) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + (2.0 / z)) / t
	t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z)
	t_3 = (x / y) + (2.0 / t)
	t_4 = (x / y) + -2.0
	tmp = 0
	if t_2 <= -2e+168:
		tmp = t_1
	elif t_2 <= -10000000000000.0:
		tmp = t_3
	elif t_2 <= -1.0:
		tmp = t_4
	elif t_2 <= 6e+209:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(2.0 / z)) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(t * z))
	t_3 = Float64(Float64(x / y) + Float64(2.0 / t))
	t_4 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -2e+168)
		tmp = t_1;
	elseif (t_2 <= -10000000000000.0)
		tmp = t_3;
	elseif (t_2 <= -1.0)
		tmp = t_4;
	elseif (t_2 <= 6e+209)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + (2.0 / z)) / t;
	t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z);
	t_3 = (x / y) + (2.0 / t);
	t_4 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_2 <= -2e+168)
		tmp = t_1;
	elseif (t_2 <= -10000000000000.0)
		tmp = t_3;
	elseif (t_2 <= -1.0)
		tmp = t_4;
	elseif (t_2 <= 6e+209)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+168], t$95$1, If[LessEqual[t$95$2, -10000000000000.0], t$95$3, If[LessEqual[t$95$2, -1.0], t$95$4, If[LessEqual[t$95$2, 6e+209], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$4]]]]]]]]]

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)) < -1.9999999999999999e168 or 5.99999999999999971e209 < (/.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 96.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 93.7%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. /-lowering-/.f64 N/A

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

   8. Simplified 93.9%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 92.7

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

   11. Simplified 92.7%

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

   if -1.9999999999999999e168 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1e13 or -1 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5.99999999999999971e209

   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

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 78.3

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

   8. Simplified 78.3%

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

   if -1e13 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1 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 60.4%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 97.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -10000000000000:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -1:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 6 \cdot 10^{+209}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_4 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -10000000000000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 6 \cdot 10^{+209}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* t z)))
        (t_2 (+ (/ x y) -2.0))
        (t_3 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* t z)))
        (t_4 (+ (/ x y) (/ 2.0 t))))
   (if (<= t_3 -2e+168)
     t_1
     (if (<= t_3 -10000000000000.0)
       t_4
       (if (<= t_3 -1.0)
         t_2
         (if (<= t_3 6e+209) t_4 (if (<= t_3 INFINITY) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, 2.0) / (t * z);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double t_4 = (x / y) + (2.0 / t);
	double tmp;
	if (t_3 <= -2e+168) {
		tmp = t_1;
	} else if (t_3 <= -10000000000000.0) {
		tmp = t_4;
	} else if (t_3 <= -1.0) {
		tmp = t_2;
	} else if (t_3 <= 6e+209) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, 2.0) / Float64(t * z))
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(t * z))
	t_4 = Float64(Float64(x / y) + Float64(2.0 / t))
	tmp = 0.0
	if (t_3 <= -2e+168)
		tmp = t_1;
	elseif (t_3 <= -10000000000000.0)
		tmp = t_4;
	elseif (t_3 <= -1.0)
		tmp = t_2;
	elseif (t_3 <= 6e+209)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + 2.0), $MachinePrecision] / 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[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+168], t$95$1, If[LessEqual[t$95$3, -10000000000000.0], t$95$4, If[LessEqual[t$95$3, -1.0], t$95$2, If[LessEqual[t$95$3, 6e+209], t$95$4, 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)) < -1.9999999999999999e168 or 5.99999999999999971e209 < (/.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 96.1%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 92.5%

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

   if -1.9999999999999999e168 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1e13 or -1 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5.99999999999999971e209

   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

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 78.3

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

   8. Simplified 78.3%

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

   if -1e13 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1 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 60.4%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 97.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -10000000000000:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -1:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 6 \cdot 10^{+209}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+238}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (* 2.0 z) (- 1.0 t))) (* t z))))
   (if (<= t_3 -1e+165)
     t_1
     (if (<= t_3 5e+199)
       t_2
       (if (<= t_3 5e+238)
         (+ -2.0 (/ 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 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -1e+165) {
		tmp = t_1;
	} else if (t_3 <= 5e+199) {
		tmp = t_2;
	} else if (t_3 <= 5e+238) {
		tmp = -2.0 + (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 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -1e+165) {
		tmp = t_1;
	} else if (t_3 <= 5e+199) {
		tmp = t_2;
	} else if (t_3 <= 5e+238) {
		tmp = -2.0 + (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 + ((2.0 * z) * (1.0 - t))) / (t * z)
	tmp = 0
	if t_3 <= -1e+165:
		tmp = t_1
	elif t_3 <= 5e+199:
		tmp = t_2
	elif t_3 <= 5e+238:
		tmp = -2.0 + (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(2.0 * z) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_3 <= -1e+165)
		tmp = t_1;
	elseif (t_3 <= 5e+199)
		tmp = t_2;
	elseif (t_3 <= 5e+238)
		tmp = Float64(-2.0 + 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 + ((2.0 * z) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if (t_3 <= -1e+165)
		tmp = t_1;
	elseif (t_3 <= 5e+199)
		tmp = t_2;
	elseif (t_3 <= 5e+238)
		tmp = -2.0 + (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[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+165], t$95$1, If[LessEqual[t$95$3, 5e+199], t$95$2, If[LessEqual[t$95$3, 5e+238], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $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)) < -9.99999999999999899e164 or 4.99999999999999995e238 < (/.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 95.7%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 66.6

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

   5. Simplified 66.6%

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

   if -9.99999999999999899e164 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.9999999999999998e199 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 78.8%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 75.0%

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

   if 4.9999999999999998e199 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.99999999999999995e238

   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 x around 0

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

   5. Simplified 90.5%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 68.6

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

   8. Simplified 68.6%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -1 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 5 \cdot 10^{+238}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* t z)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* t z)))
        (t_3 (+ (/ x y) -2.0)))
   (if (<= t_2 -1e+81)
     t_1
     (if (<= t_2 20.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, 2.0) / (t * z);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -1e+81) {
		tmp = t_1;
	} else if (t_2 <= 20.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, 2.0) / Float64(t * z))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(t * z))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -1e+81)
		tmp = t_1;
	elseif (t_2 <= 20.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+81], t$95$1, If[LessEqual[t$95$2, 20.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

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)) < -9.99999999999999921e80 or 20 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.8%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 76.1%

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

   if -9.99999999999999921e80 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 20 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 66.0%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 94.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 20:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(t, -2, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6.1e+156)
   (+ (/ x y) (/ 2.0 t))
   (if (<= (/ x y) 4.2e-7)
     (/ (+ 2.0 (fma t -2.0 (/ 2.0 z))) t)
     (+ (/ x y) (+ -2.0 (/ 2.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6.1e+156) {
		tmp = (x / y) + (2.0 / t);
	} else if ((x / y) <= 4.2e-7) {
		tmp = (2.0 + fma(t, -2.0, (2.0 / z))) / t;
	} else {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6.1e+156)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (Float64(x / y) <= 4.2e-7)
		tmp = Float64(Float64(2.0 + fma(t, -2.0, Float64(2.0 / z))) / t);
	else
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -6.1e+156], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 4.2e-7], N[(N[(2.0 + N[(t * -2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -6.1000000000000001e156

   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 t around 0

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 87.5

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

   8. Simplified 87.5%

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

   if -6.1000000000000001e156 < (/.f64 x y) < 4.2e-7

   1. Initial program 85.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 94.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. /-lowering-/.f64 N/A

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

   8. Simplified 94.3%

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

   if 4.2e-7 < (/.f64 x y)

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 88.7

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

   5. Simplified 88.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(t, -2, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6.1e+156)
   (+ (/ x y) (/ 2.0 t))
   (if (<= (/ x y) 4.8e-7)
     (fma (/ 2.0 (* t z)) (+ z 1.0) -2.0)
     (+ (/ x y) (+ -2.0 (/ 2.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6.1e+156) {
		tmp = (x / y) + (2.0 / t);
	} else if ((x / y) <= 4.8e-7) {
		tmp = fma((2.0 / (t * z)), (z + 1.0), -2.0);
	} else {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6.1e+156)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (Float64(x / y) <= 4.8e-7)
		tmp = fma(Float64(2.0 / Float64(t * z)), Float64(z + 1.0), -2.0);
	else
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -6.1e+156], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 4.8e-7], N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -6.1000000000000001e156

   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 t around 0

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 87.5

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

   8. Simplified 87.5%

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

   if -6.1000000000000001e156 < (/.f64 x y) < 4.79999999999999957e-7

   1. Initial program 85.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 94.0%

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

   if 4.79999999999999957e-7 < (/.f64 x y)

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 88.7

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

   5. Simplified 88.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -6.4 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 7.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t \cdot z}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 t))))
   (if (<= (/ x y) -6.4e+156)
     t_1
     (if (<= (/ x y) 7.5) (fma (/ 2.0 (* t z)) (+ z 1.0) -2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / t);
	double tmp;
	if ((x / y) <= -6.4e+156) {
		tmp = t_1;
	} else if ((x / y) <= 7.5) {
		tmp = fma((2.0 / (t * z)), (z + 1.0), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / t))
	tmp = 0.0
	if (Float64(x / y) <= -6.4e+156)
		tmp = t_1;
	elseif (Float64(x / y) <= 7.5)
		tmp = fma(Float64(2.0 / Float64(t * z)), Float64(z + 1.0), -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -6.4e+156], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 7.5], N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -6.40000000000000005e156 or 7.5 < (/.f64 x y)

   1. Initial program 82.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 96.7%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 86.6

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

   8. Simplified 86.6%

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

   if -6.40000000000000005e156 < (/.f64 x y) < 7.5

   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

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

   5. Simplified 93.3%

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

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t \cdot z} + -2\right)\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (+ (/ 2.0 (* t z)) -2.0))))
   (if (<= t -2.6e+21)
     t_1
     (if (<= t 2.8e-6) (+ (/ x y) (/ (fma 2.0 z 2.0) (* t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + ((2.0 / (t * z)) + -2.0);
	double tmp;
	if (t <= -2.6e+21) {
		tmp = t_1;
	} else if (t <= 2.8e-6) {
		tmp = (x / y) + (fma(2.0, z, 2.0) / (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(Float64(2.0 / Float64(t * z)) + -2.0))
	tmp = 0.0
	if (t <= -2.6e+21)
		tmp = t_1;
	elseif (t <= 2.8e-6)
		tmp = Float64(Float64(x / y) + Float64(fma(2.0, z, 2.0) / Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.6e+21], t$95$1, If[LessEqual[t, 2.8e-6], N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e21 or 2.79999999999999987e-6 < t

   1. Initial program 68.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 98.8%

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

      1. +-lowering-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 98.8

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

   10. Applied egg-rr 98.8%

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

   if -2.6e21 < t < 2.79999999999999987e-6

   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

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

      1. associate-/r* N/A

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

      2. remove-double-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-frac N/A

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

      11. associate-/r* N/A

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

      12. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

Alternative 10: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -1.95 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= (/ x y) -1.95e-6)
     t_1
     (if (<= (/ x y) 3.8e-7) (+ -2.0 (/ 2.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if ((x / y) <= -1.95e-6) {
		tmp = t_1;
	} else if ((x / y) <= 3.8e-7) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) + (-2.0d0)
    if ((x / y) <= (-1.95d-6)) then
        tmp = t_1
    else if ((x / y) <= 3.8d-7) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if ((x / y) <= -1.95e-6) {
		tmp = t_1;
	} else if ((x / y) <= 3.8e-7) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) + -2.0
	tmp = 0
	if (x / y) <= -1.95e-6:
		tmp = t_1
	elif (x / y) <= 3.8e-7:
		tmp = -2.0 + (2.0 / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (Float64(x / y) <= -1.95e-6)
		tmp = t_1;
	elseif (Float64(x / y) <= 3.8e-7)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) + -2.0;
	tmp = 0.0;
	if ((x / y) <= -1.95e-6)
		tmp = t_1;
	elseif ((x / y) <= 3.8e-7)
		tmp = -2.0 + (2.0 / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1.95e-6], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 3.8e-7], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.95e-6 or 3.80000000000000015e-7 < (/.f64 x y)

   1. Initial program 84.1%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 71.3%

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

   if -1.95e-6 < (/.f64 x y) < 3.80000000000000015e-7

   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

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 65.7

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

   8. Simplified 65.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.95 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -48000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5.1 \cdot 10^{+53}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -48000.0)
   (/ x y)
   (if (<= (/ x y) 5.1e+53) (+ -2.0 (/ 2.0 t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -48000.0) {
		tmp = x / y;
	} else if ((x / y) <= 5.1e+53) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x / y) <= (-48000.0d0)) then
        tmp = x / y
    else if ((x / y) <= 5.1d+53) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -48000.0) {
		tmp = x / y;
	} else if ((x / y) <= 5.1e+53) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -48000.0:
		tmp = x / y
	elif (x / y) <= 5.1e+53:
		tmp = -2.0 + (2.0 / t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -48000.0)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 5.1e+53)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -48000.0)
		tmp = x / y;
	elseif ((x / y) <= 5.1e+53)
		tmp = -2.0 + (2.0 / t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -48000.0], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5.1e+53], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -48000 or 5.0999999999999998e53 < (/.f64 x y)

   1. Initial program 85.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 71.9

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

   5. Simplified 71.9%

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

   if -48000 < (/.f64 x y) < 5.0999999999999998e53

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 95.8%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 63.7

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

   8. Simplified 63.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -48000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5.1 \cdot 10^{+53}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t \cdot z} + -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (+ -2.0 (/ 2.0 t)))))
   (if (<= z -1.0)
     t_1
     (if (<= z 1.9e-7) (+ (/ x y) (+ (/ 2.0 (* t z)) -2.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (-2.0 + (2.0 / t));
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.9e-7) {
		tmp = (x / y) + ((2.0 / (t * z)) + -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) + ((-2.0d0) + (2.0d0 / t))
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 1.9d-7) then
        tmp = (x / y) + ((2.0d0 / (t * z)) + (-2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (-2.0 + (2.0 / t));
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.9e-7) {
		tmp = (x / y) + ((2.0 / (t * z)) + -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) + (-2.0 + (2.0 / t))
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 1.9e-7:
		tmp = (x / y) + ((2.0 / (t * z)) + -2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.9e-7)
		tmp = Float64(Float64(x / y) + Float64(Float64(2.0 / Float64(t * z)) + -2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) + (-2.0 + (2.0 / t));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.9e-7)
		tmp = (x / y) + ((2.0 / (t * z)) + -2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 1.9e-7], N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.90000000000000007e-7 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if -1 < z < 1.90000000000000007e-7

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 97.6%

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

      1. +-lowering-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 97.6

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t \cdot z} + -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 70.1

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

   5. Simplified 70.1%

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

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

   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

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 36.0%

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

Alternative 14: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;z \leq -8 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;-2 + \frac{2}{\mathsf{fma}\left(t, z, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= z -8e+137)
     t_1
     (if (<= z -3.6e-15)
       (+ -2.0 (/ 2.0 t))
       (if (<= z 1.9e-58) (+ -2.0 (/ 2.0 (fma t z 0.0))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if (z <= -8e+137) {
		tmp = t_1;
	} else if (z <= -3.6e-15) {
		tmp = -2.0 + (2.0 / t);
	} else if (z <= 1.9e-58) {
		tmp = -2.0 + (2.0 / fma(t, z, 0.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (z <= -8e+137)
		tmp = t_1;
	elseif (z <= -3.6e-15)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	elseif (z <= 1.9e-58)
		tmp = Float64(-2.0 + Float64(2.0 / fma(t, z, 0.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[z, -8e+137], t$95$1, If[LessEqual[z, -3.6e-15], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e-58], N[(-2.0 + N[(2.0 / N[(t * z + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000003e137 or 1.8999999999999999e-58 < z

   1. Initial program 72.5%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 71.2%

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

   if -8.0000000000000003e137 < z < -3.6000000000000001e-15

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 70.3%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 64.8

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

   8. Simplified 64.8%

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

   if -3.6000000000000001e-15 < z < 1.8999999999999999e-58

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 97.8%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. /-lowering-/.f64 N/A

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

       8. remove-double-neg N/A

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

       9. +-rgt-identity N/A

          \[\leadsto -2 + \frac{2}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t \cdot z\right)\right) + 0\right)}\right)} \]

       10. distribute-neg-in N/A

           \[\leadsto -2 + \frac{2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t \cdot z\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}} \]

       11. remove-double-neg N/A

           \[\leadsto -2 + \frac{2}{\color{blue}{t \cdot z} + \left(\mathsf{neg}\left(0\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto -2 + \frac{2}{t \cdot z + \color{blue}{0}} \]

       13. accelerator-lowering-fma.f64 79.0

           \[\leadsto -2 + \frac{2}{\color{blue}{\mathsf{fma}\left(t, z, 0\right)}} \]

   11. Simplified 79.0%

       \[\leadsto \color{blue}{-2 + \frac{2}{\mathsf{fma}\left(t, z, 0\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;-2 + \frac{2}{\mathsf{fma}\left(t, z, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+21}:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e+21) -2.0 (if (<= t 1.0) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+21) {
		tmp = -2.0;
	} else if (t <= 1.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.6d+21)) then
        tmp = -2.0d0
    else if (t <= 1.0d0) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+21) {
		tmp = -2.0;
	} else if (t <= 1.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.6e+21:
		tmp = -2.0
	elif t <= 1.0:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e+21)
		tmp = -2.0;
	elseif (t <= 1.0)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.6e+21)
		tmp = -2.0;
	elseif (t <= 1.0)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e+21], -2.0, If[LessEqual[t, 1.0], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e21 or 1 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 54.0%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 39.3%

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

   if -2.6e21 < t < 1

   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 z around 0

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in z around inf

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

       1. /-lowering-/.f64 35.8

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

   11. Simplified 35.8%

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

Alternative 16: 20.6% 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 84.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 65.2%

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

6. Taylor expanded in t around inf

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

8. Simplified 19.0%

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

Developer Target 1: 99.2% 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 2024195 
(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))))