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

Percentage Accurate: 86.5% → 99.3%

Time: 13.1s

Alternatives: 14

Speedup: 1.2×

• 23.8% 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: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in 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 99.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t))))
   (if (<= t_2 -5e+44)
     t_1
     (if (<= t_2 4e+23)
       (/ (fma y -2.0 x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, 2.0) / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double tmp;
	if (t_2 <= -5e+44) {
		tmp = t_1;
	} else if (t_2 <= 4e+23) {
		tmp = fma(y, -2.0, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, 2.0) / Float64(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+44)
		tmp = t_1;
	elseif (t_2 <= 4e+23)
		tmp = Float64(fma(y, -2.0, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+44], t$95$1, If[LessEqual[t$95$2, 4e+23], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999996e44 or 3.9999999999999997e23 < (/.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 80.9%

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

   if -4.9999999999999996e44 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 3.9999999999999997e23

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-add N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. *-inverses N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 92.5

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

   7. Applied rewrites 92.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -110000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 3.4 \cdot 10^{-176}:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{x}{y} \leq 8200000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -110000000000.0)
   (/ x y)
   (if (<= (/ x y) 3.4e-176)
     -2.0
     (if (<= (/ x y) 8200000.0) (/ 2.0 t) (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -110000000000.0) {
		tmp = x / y;
	} else if ((x / y) <= 3.4e-176) {
		tmp = -2.0;
	} else if ((x / y) <= 8200000.0) {
		tmp = 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) <= (-110000000000.0d0)) then
        tmp = x / y
    else if ((x / y) <= 3.4d-176) then
        tmp = -2.0d0
    else if ((x / y) <= 8200000.0d0) then
        tmp = 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) <= -110000000000.0) {
		tmp = x / y;
	} else if ((x / y) <= 3.4e-176) {
		tmp = -2.0;
	} else if ((x / y) <= 8200000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -110000000000.0:
		tmp = x / y
	elif (x / y) <= 3.4e-176:
		tmp = -2.0
	elif (x / y) <= 8200000.0:
		tmp = 2.0 / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -110000000000.0)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 3.4e-176)
		tmp = -2.0;
	elseif (Float64(x / y) <= 8200000.0)
		tmp = 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) <= -110000000000.0)
		tmp = x / y;
	elseif ((x / y) <= 3.4e-176)
		tmp = -2.0;
	elseif ((x / y) <= 8200000.0)
		tmp = 2.0 / t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -110000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 3.4e-176], -2.0, If[LessEqual[N[(x / y), $MachinePrecision], 8200000.0], N[(2.0 / t), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.1e11 or 8.2e6 < (/.f64 x y)

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.1e11 < (/.f64 x y) < 3.3999999999999997e-176

   1. Initial program 93.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 85.9%

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

   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 98.6%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 35.8%

       \[\leadsto -2 \]

   if 3.3999999999999997e-176 < (/.f64 x y) < 8.2e6

   1. Initial program 90.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 46.1%

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

Alternative 4: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e+81)
   (+ (/ x y) (/ 2.0 t))
   (if (<= (/ x y) 200.0)
     (fma (/ 2.0 (* z t)) (+ 1.0 z) -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) <= -5e+81) {
		tmp = (x / y) + (2.0 / t);
	} else if ((x / y) <= 200.0) {
		tmp = fma((2.0 / (z * t)), (1.0 + z), -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) <= -5e+81)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (Float64(x / y) <= 200.0)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(1.0 + z), -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], -5e+81], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 200.0], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(1.0 + z), $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) < -4.9999999999999998e81

   1. Initial program 81.2%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 80.1%

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

   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 86.8

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.8%

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

   if -4.9999999999999998e81 < (/.f64 x y) < 200

   1. Initial program 93.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 95.9%

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

   if 200 < (/.f64 x y)

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 t))))
   (if (<= (/ x y) -5e+81)
     t_1
     (if (<= (/ x y) 200.0) (fma (/ 2.0 (* z t)) (+ 1.0 z) -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+81) {
		tmp = t_1;
	} else if ((x / y) <= 200.0) {
		tmp = fma((2.0 / (z * t)), (1.0 + 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 / t))
	tmp = 0.0
	if (Float64(x / y) <= -5e+81)
		tmp = t_1;
	elseif (Float64(x / y) <= 200.0)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(1.0 + z), -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+81], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 200.0], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(1.0 + z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4.9999999999999998e81 or 200 < (/.f64 x y)

   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. 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 74.0%

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

   5. Taylor expanded in 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 90.3

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 89.6%

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

   if -4.9999999999999998e81 < (/.f64 x y) < 200

   1. Initial program 93.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 95.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 40000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -4e+159)
   (/ x y)
   (if (<= (/ x y) 40000.0)
     (fma (/ 2.0 (* z t)) (+ 1.0 z) -2.0)
     (/ (fma y -2.0 x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -4e+159) {
		tmp = x / y;
	} else if ((x / y) <= 40000.0) {
		tmp = fma((2.0 / (z * t)), (1.0 + z), -2.0);
	} else {
		tmp = fma(y, -2.0, x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -4e+159)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 40000.0)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(1.0 + z), -2.0);
	else
		tmp = Float64(fma(y, -2.0, x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -4e+159], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 40000.0], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(1.0 + z), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(y * -2.0 + x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -3.9999999999999997e159

   1. Initial program 78.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -3.9999999999999997e159 < (/.f64 x y) < 4e4

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.2%

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

   if 4e4 < (/.f64 x y)

   1. Initial program 87.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-add N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. *-inverses N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 83.3

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

   7. Applied rewrites 83.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 40000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, 1 + z, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -2, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 40000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= (/ x y) -4e-14)
     t_1
     (if (<= (/ x y) 40000.0) (+ -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) <= -4e-14) {
		tmp = t_1;
	} else if ((x / y) <= 40000.0) {
		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) <= (-4d-14)) then
        tmp = t_1
    else if ((x / y) <= 40000.0d0) 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) <= -4e-14) {
		tmp = t_1;
	} else if ((x / y) <= 40000.0) {
		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) <= -4e-14:
		tmp = t_1
	elif (x / y) <= 40000.0:
		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) <= -4e-14)
		tmp = t_1;
	elseif (Float64(x / y) <= 40000.0)
		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) <= -4e-14)
		tmp = t_1;
	elseif ((x / y) <= 40000.0)
		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], -4e-14], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 40000.0], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4e-14 or 4e4 < (/.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 73.7%

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

   if -4e-14 < (/.f64 x y) < 4e4

   1. Initial program 92.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 80.1%

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

   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 99.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 40000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8200000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1.02e+27)
   (/ x y)
   (if (<= (/ x y) 8200000.0) (+ -2.0 (/ 2.0 t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1.02e+27) {
		tmp = x / y;
	} else if ((x / y) <= 8200000.0) {
		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) <= (-1.02d+27)) then
        tmp = x / y
    else if ((x / y) <= 8200000.0d0) 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) <= -1.02e+27) {
		tmp = x / y;
	} else if ((x / y) <= 8200000.0) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -1.02e+27:
		tmp = x / y
	elif (x / y) <= 8200000.0:
		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) <= -1.02e+27)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 8200000.0)
		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) <= -1.02e+27)
		tmp = x / y;
	elseif ((x / y) <= 8200000.0)
		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], -1.02e+27], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 8200000.0], 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) < -1.0199999999999999e27 or 8.2e6 < (/.f64 x y)

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -1.0199999999999999e27 < (/.f64 x y) < 8.2e6

   1. Initial program 93.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 80.9%

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

   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 98.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 58.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8200000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (+ -2.0 (/ 2.0 t)))))
   (if (<= z -9.2e-7)
     t_1
     (if (<= z 7.2e-70) (+ (/ x y) (/ 2.0 (* z t))) 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 <= -9.2e-7) {
		tmp = t_1;
	} else if (z <= 7.2e-70) {
		tmp = (x / y) + (2.0 / (z * 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) + (2.0d0 / t))
    if (z <= (-9.2d-7)) then
        tmp = t_1
    else if (z <= 7.2d-70) then
        tmp = (x / y) + (2.0d0 / (z * 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 + (2.0 / t));
	double tmp;
	if (z <= -9.2e-7) {
		tmp = t_1;
	} else if (z <= 7.2e-70) {
		tmp = (x / y) + (2.0 / (z * t));
	} 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 <= -9.2e-7:
		tmp = t_1
	elif z <= 7.2e-70:
		tmp = (x / y) + (2.0 / (z * t))
	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 <= -9.2e-7)
		tmp = t_1;
	elseif (z <= 7.2e-70)
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(z * t)));
	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 <= -9.2e-7)
		tmp = t_1;
	elseif (z <= 7.2e-70)
		tmp = (x / y) + (2.0 / (z * 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] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e-7], t$95$1, If[LessEqual[z, 7.2e-70], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999998e-7 or 7.2000000000000004e-70 < z

   1. Initial program 84.3%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -9.1999999999999998e-7 < z < 7.2000000000000004e-70

   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

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

   5. Applied rewrites 89.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in 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 99.0%

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

6. Final simplification 99.0%

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

Alternative 11: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+239}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= z -1.85e+239)
     (+ -2.0 (/ 2.0 t))
     (if (<= z -1.32e-121) t_1 (if (<= z 1.9e-73) (/ 2.0 (* z t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if (z <= -1.85e+239) {
		tmp = -2.0 + (2.0 / t);
	} else if (z <= -1.32e-121) {
		tmp = t_1;
	} else if (z <= 1.9e-73) {
		tmp = 2.0 / (z * 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 (z <= (-1.85d+239)) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else if (z <= (-1.32d-121)) then
        tmp = t_1
    else if (z <= 1.9d-73) then
        tmp = 2.0d0 / (z * 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 (z <= -1.85e+239) {
		tmp = -2.0 + (2.0 / t);
	} else if (z <= -1.32e-121) {
		tmp = t_1;
	} else if (z <= 1.9e-73) {
		tmp = 2.0 / (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) + -2.0
	tmp = 0
	if z <= -1.85e+239:
		tmp = -2.0 + (2.0 / t)
	elif z <= -1.32e-121:
		tmp = t_1
	elif z <= 1.9e-73:
		tmp = 2.0 / (z * 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 (z <= -1.85e+239)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	elseif (z <= -1.32e-121)
		tmp = t_1;
	elseif (z <= 1.9e-73)
		tmp = Float64(2.0 / Float64(z * 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 (z <= -1.85e+239)
		tmp = -2.0 + (2.0 / t);
	elseif (z <= -1.32e-121)
		tmp = t_1;
	elseif (z <= 1.9e-73)
		tmp = 2.0 / (z * 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[z, -1.85e+239], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.32e-121], t$95$1, If[LessEqual[z, 1.9e-73], N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999999e239

   1. Initial program 76.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 19.0%

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

   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 77.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 77.7%

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

   if -1.84999999999999999e239 < z < -1.32e-121 or 1.9000000000000001e-73 < z

   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. Taylor expanded in t around inf

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

   5. Applied rewrites 69.1%

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

   if -1.32e-121 < z < 1.9000000000000001e-73

   1. Initial program 97.3%

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

   3. Taylor expanded in 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 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+239}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-121}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;t \leq -4.15 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{2 \cdot z}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) -2.0)))
   (if (<= t -4.15e-81) t_1 (if (<= t 3.2e-48) (/ (* 2.0 z) (* z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + -2.0;
	double tmp;
	if (t <= -4.15e-81) {
		tmp = t_1;
	} else if (t <= 3.2e-48) {
		tmp = (2.0 * z) / (z * 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 (t <= (-4.15d-81)) then
        tmp = t_1
    else if (t <= 3.2d-48) then
        tmp = (2.0d0 * z) / (z * 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 (t <= -4.15e-81) {
		tmp = t_1;
	} else if (t <= 3.2e-48) {
		tmp = (2.0 * z) / (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) + -2.0
	tmp = 0
	if t <= -4.15e-81:
		tmp = t_1
	elif t <= 3.2e-48:
		tmp = (2.0 * z) / (z * 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 (t <= -4.15e-81)
		tmp = t_1;
	elseif (t <= 3.2e-48)
		tmp = Float64(Float64(2.0 * z) / Float64(z * 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 (t <= -4.15e-81)
		tmp = t_1;
	elseif (t <= 3.2e-48)
		tmp = (2.0 * z) / (z * 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[t, -4.15e-81], t$95$1, If[LessEqual[t, 3.2e-48], N[(N[(2.0 * z), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.15000000000000007e-81 or 3.1999999999999998e-48 < t

   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 t around inf

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

   5. Applied rewrites 75.3%

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

   if -4.15000000000000007e-81 < t < 3.1999999999999998e-48

   1. Initial program 97.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 61.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{2 \cdot z}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.0) -2.0 (if (<= t 5.5e-12) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.0) {
		tmp = -2.0;
	} else if (t <= 5.5e-12) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t <= 5.5d-12) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.0) {
		tmp = -2.0;
	} else if (t <= 5.5e-12) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.0:
		tmp = -2.0
	elif t <= 5.5e-12:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.0)
		tmp = -2.0;
	elseif (t <= 5.5e-12)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.0)
		tmp = -2.0;
	elseif (t <= 5.5e-12)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.0], -2.0, If[LessEqual[t, 5.5e-12], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -1 or 5.5000000000000004e-12 < t

   1. Initial program 81.8%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 85.2%

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

   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 54.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 34.9%

       \[\leadsto -2 \]

   if -1 < t < 5.5000000000000004e-12

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.9%

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

Alternative 14: 20.0% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
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 78.2%

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

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 64.7%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 19.3%

    \[\leadsto -2 \]
11. Add Preprocessing

Developer Target 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(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))))