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

Percentage Accurate: 86.6% → 99.1%

Time: 13.3s

Alternatives: 14

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: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in 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. Applied rewrites 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (/ 2.0 z)) t))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -1e+95)
     t_1
     (if (<= t_2 4e+149)
       (+ (/ x y) (+ -2.0 (/ 2.0 t)))
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -1e+95) {
		tmp = t_1;
	} else if (t_2 <= 4e+149) {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -1e+95) {
		tmp = t_1;
	} else if (t_2 <= 4e+149) {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	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))) / (z * t)
	tmp = 0
	if t_2 <= -1e+95:
		tmp = t_1
	elif t_2 <= 4e+149:
		tmp = (x / y) + (-2.0 + (2.0 / t))
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = (x / y) + -2.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(2.0 / z)) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -1e+95)
		tmp = t_1;
	elseif (t_2 <= 4e+149)
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + (2.0 / z)) / t;
	t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	tmp = 0.0;
	if (t_2 <= -1e+95)
		tmp = t_1;
	elseif (t_2 <= 4e+149)
		tmp = (x / y) + (-2.0 + (2.0 / t));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = (x / y) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+95], t$95$1, If[LessEqual[t$95$2, 4e+149], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.00000000000000002e95 or 4.0000000000000002e149 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 89.9%

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

   if -1.00000000000000002e95 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.0000000000000002e149

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 92.4

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

   5. Applied rewrites 92.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (/ 2.0 z)) t))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t)))
        (t_3 (+ (/ x y) -2.0)))
   (if (<= t_2 -5e+49)
     t_1
     (if (<= t_2 2e+29) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -5e+49) {
		tmp = t_1;
	} else if (t_2 <= 2e+29) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -5e+49) {
		tmp = t_1;
	} else if (t_2 <= 2e+29) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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))) / (z * t)
	t_3 = (x / y) + -2.0
	tmp = 0
	if t_2 <= -5e+49:
		tmp = t_1
	elif t_2 <= 2e+29:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(2.0 / z)) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -5e+49)
		tmp = t_1;
	elseif (t_2 <= 2e+29)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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))) / (z * t);
	t_3 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_2 <= -5e+49)
		tmp = t_1;
	elseif (t_2 <= 2e+29)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	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[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+49], t$95$1, If[LessEqual[t$95$2, 2e+29], 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)) < -5.0000000000000004e49 or 1.99999999999999983e29 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 81.8%

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

   if -5.0000000000000004e49 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.99999999999999983e29 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 74.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ t_2 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{2}{z \cdot t} \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t)))
        (t_2 (+ (/ x y) -2.0)))
   (if (<= t_1 -5e+49)
     (/ (fma 2.0 z 2.0) (* z t))
     (if (<= t_1 2e+29)
       t_2
       (if (<= t_1 INFINITY) (* (/ 2.0 (* z t)) (+ z 1.0)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_2 = (x / y) + -2.0;
	double tmp;
	if (t_1 <= -5e+49) {
		tmp = fma(2.0, z, 2.0) / (z * t);
	} else if (t_1 <= 2e+29) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (2.0 / (z * t)) * (z + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	t_2 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_1 <= -5e+49)
		tmp = Float64(fma(2.0, z, 2.0) / Float64(z * t));
	elseif (t_1 <= 2e+29)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(2.0 / Float64(z * t)) * Float64(z + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+49], N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+29], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5.0000000000000004e49

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 86.9%

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

   if -5.0000000000000004e49 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.99999999999999983e29 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 74.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.0%

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

   if 1.99999999999999983e29 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2}{z \cdot t} \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t)))
        (t_3 (+ (/ x y) -2.0)))
   (if (<= t_2 -5e+49)
     t_1
     (if (<= t_2 2e+29) 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) / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -5e+49) {
		tmp = t_1;
	} else if (t_2 <= 2e+29) {
		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(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -5e+49)
		tmp = t_1;
	elseif (t_2 <= 2e+29)
		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[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+49], t$95$1, If[LessEqual[t$95$2, 2e+29], 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)) < -5.0000000000000004e49 or 1.99999999999999983e29 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 81.5%

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

   if -5.0000000000000004e49 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.99999999999999983e29 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 74.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t)))
        (t_2 (+ (/ x y) -2.0))
        (t_3 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_3 -1e+95)
     t_1
     (if (<= t_3 4e+149) t_2 (if (<= t_3 INFINITY) t_1 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_3 <= -1e+95) {
		tmp = t_1;
	} else if (t_3 <= 4e+149) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_3 <= -1e+95) {
		tmp = t_1;
	} else if (t_3 <= 4e+149) {
		tmp = t_2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (z * t)
	t_2 = (x / y) + -2.0
	t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t)
	tmp = 0
	if t_3 <= -1e+95:
		tmp = t_1
	elif t_3 <= 4e+149:
		tmp = t_2
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_3 <= -1e+95)
		tmp = t_1;
	elseif (t_3 <= 4e+149)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (z * t);
	t_2 = (x / y) + -2.0;
	t_3 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	tmp = 0.0;
	if (t_3 <= -1e+95)
		tmp = t_1;
	elseif (t_3 <= 4e+149)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+95], t$95$1, If[LessEqual[t$95$3, 4e+149], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.00000000000000002e95 or 4.0000000000000002e149 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if -1.00000000000000002e95 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.0000000000000002e149 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 79.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 88.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ (fma 2.0 z 2.0) (* z t)))))
   (if (<= (/ x y) -50000.0)
     t_1
     (if (<= (/ x y) 2e-18) (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (fma(2.0, z, 2.0) / (z * t));
	double tmp;
	if ((x / y) <= -50000.0) {
		tmp = t_1;
	} else if ((x / y) <= 2e-18) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(fma(2.0, z, 2.0) / Float64(z * t)))
	tmp = 0.0
	if (Float64(x / y) <= -50000.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 2e-18)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -50000.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 2e-18], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5e4 or 2.0000000000000001e-18 < (/.f64 x y)

   1. Initial program 87.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -5e4 < (/.f64 x y) < 2.0000000000000001e-18

   1. Initial program 86.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -50000:\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t))))
   (if (<= (/ x y) -5e+14)
     (+ (/ x y) (+ -2.0 (/ 2.0 t)))
     (if (<= (/ x y) 2e-18) (fma t_1 (+ z 1.0) -2.0) (+ (/ x y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double tmp;
	if ((x / y) <= -5e+14) {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	} else if ((x / y) <= 2e-18) {
		tmp = fma(t_1, (z + 1.0), -2.0);
	} else {
		tmp = (x / y) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	tmp = 0.0
	if (Float64(x / y) <= -5e+14)
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	elseif (Float64(x / y) <= 2e-18)
		tmp = fma(t_1, Float64(z + 1.0), -2.0);
	else
		tmp = Float64(Float64(x / y) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+14], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e-18], N[(t$95$1 * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -5e14

   1. Initial program 81.7%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -5e14 < (/.f64 x y) < 2.0000000000000001e-18

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.0%

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

   if 2.0000000000000001e-18 < (/.f64 x y)

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 90.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 t))))
   (if (<= (/ x y) -5e+14)
     t_1
     (if (<= (/ x y) 20000.0) (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / t);
	double tmp;
	if ((x / y) <= -5e+14) {
		tmp = t_1;
	} else if ((x / y) <= 20000.0) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / t))
	tmp = 0.0
	if (Float64(x / y) <= -5e+14)
		tmp = t_1;
	elseif (Float64(x / y) <= 20000.0)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+14], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 20000.0], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5e14 or 2e4 < (/.f64 x y)

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in z around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-in N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. sub-neg N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-in N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 79.3

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.7%

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

   if -5e14 < (/.f64 x y) < 2e4

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -5800:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 11200:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= (/ x y) -5800.0)
     t_1
     (if (<= (/ x y) 11200.0) (+ -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) <= -5800.0) {
		tmp = t_1;
	} else if ((x / y) <= 11200.0) {
		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) <= (-5800.0d0)) then
        tmp = t_1
    else if ((x / y) <= 11200.0d0) 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) <= -5800.0) {
		tmp = t_1;
	} else if ((x / y) <= 11200.0) {
		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) <= -5800.0:
		tmp = t_1
	elif (x / y) <= 11200.0:
		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) <= -5800.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 11200.0)
		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) <= -5800.0)
		tmp = t_1;
	elseif ((x / y) <= 11200.0)
		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], -5800.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 11200.0], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5800 or 11200 < (/.f64 x y)

   1. Initial program 87.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 69.2%

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

   if -5800 < (/.f64 x y) < 11200

   1. Initial program 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 65.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5800:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 11200:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -52000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 25000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -52000.0)
   (/ x y)
   (if (<= (/ x y) 25000.0) (+ -2.0 (/ 2.0 t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -52000.0) {
		tmp = x / y;
	} else if ((x / y) <= 25000.0) {
		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) <= (-52000.0d0)) then
        tmp = x / y
    else if ((x / y) <= 25000.0d0) 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) <= -52000.0) {
		tmp = x / y;
	} else if ((x / y) <= 25000.0) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -52000.0:
		tmp = x / y
	elif (x / y) <= 25000.0:
		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) <= -52000.0)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 25000.0)
		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) <= -52000.0)
		tmp = x / y;
	elseif ((x / y) <= 25000.0)
		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], -52000.0], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 25000.0], 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) < -52000 or 25000 < (/.f64 x y)

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -52000 < (/.f64 x y) < 25000

   1. Initial program 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 65.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -52000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 25000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 11200:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6e-21) (/ x y) (if (<= (/ x y) 11200.0) -2.0 (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6e-21) {
		tmp = x / y;
	} else if ((x / y) <= 11200.0) {
		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) <= (-6d-21)) then
        tmp = x / y
    else if ((x / y) <= 11200.0d0) 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) <= -6e-21) {
		tmp = x / y;
	} else if ((x / y) <= 11200.0) {
		tmp = -2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -6e-21:
		tmp = x / y
	elif (x / y) <= 11200.0:
		tmp = -2.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6e-21)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 11200.0)
		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) <= -6e-21)
		tmp = x / y;
	elseif ((x / y) <= 11200.0)
		tmp = -2.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -6e-21], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 11200.0], -2.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5.99999999999999982e-21 or 11200 < (/.f64 x y)

   1. Initial program 87.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -5.99999999999999982e-21 < (/.f64 x y) < 11200

   1. Initial program 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 46.0%

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

Alternative 13: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e+14) -2.0 (if (<= t 2.45e+14) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e+14) {
		tmp = -2.0;
	} else if (t <= 2.45e+14) {
		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 <= (-7.5d+14)) then
        tmp = -2.0d0
    else if (t <= 2.45d+14) 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 <= -7.5e+14) {
		tmp = -2.0;
	} else if (t <= 2.45e+14) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.5e+14:
		tmp = -2.0
	elif t <= 2.45e+14:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e+14)
		tmp = -2.0;
	elseif (t <= 2.45e+14)
		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 <= -7.5e+14)
		tmp = -2.0;
	elseif (t <= 2.45e+14)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e+14], -2.0, If[LessEqual[t, 2.45e+14], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e14 or 2.45e14 < t

   1. Initial program 76.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 39.4%

       \[\leadsto -2 \]

   if -7.5e14 < t < 2.45e14

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 32.0%

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

Alternative 14: 19.8% 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 86.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 80.2%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-out N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft1-in N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 63.3%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 21.9%

    \[\leadsto -2 \]
11. 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 2024220 
(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))))