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

Percentage Accurate: 86.5% → 99.4%

Time: 10.2s

Alternatives: 12

Speedup: 0.6×

• 24.0% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

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

Alternative 2: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := -2 + \frac{x}{y}\\ t_3 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z)))
        (t_2 (+ -2.0 (/ x y)))
        (t_3 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z))))
   (if (<= t_3 -2e+148)
     t_1
     (if (<= t_3 -1.0)
       t_2
       (if (<= t_3 5e+294)
         (- (/ 2.0 t) 2.0)
         (if (<= t_3 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = -2.0 + (x / y);
	double t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double tmp;
	if (t_3 <= -2e+148) {
		tmp = t_1;
	} else if (t_3 <= -1.0) {
		tmp = t_2;
	} else if (t_3 <= 5e+294) {
		tmp = (2.0 / t) - 2.0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = -2.0 + (x / y);
	double t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double tmp;
	if (t_3 <= -2e+148) {
		tmp = t_1;
	} else if (t_3 <= -1.0) {
		tmp = t_2;
	} else if (t_3 <= 5e+294) {
		tmp = (2.0 / t) - 2.0;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (t * z)
	t_2 = -2.0 + (x / y)
	t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)
	tmp = 0
	if t_3 <= -2e+148:
		tmp = t_1
	elif t_3 <= -1.0:
		tmp = t_2
	elif t_3 <= 5e+294:
		tmp = (2.0 / t) - 2.0
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	t_2 = Float64(-2.0 + Float64(x / y))
	t_3 = Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z))
	tmp = 0.0
	if (t_3 <= -2e+148)
		tmp = t_1;
	elseif (t_3 <= -1.0)
		tmp = t_2;
	elseif (t_3 <= 5e+294)
		tmp = Float64(Float64(2.0 / t) - 2.0);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (t * z);
	t_2 = -2.0 + (x / y);
	t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	tmp = 0.0;
	if (t_3 <= -2e+148)
		tmp = t_1;
	elseif (t_3 <= -1.0)
		tmp = t_2;
	elseif (t_3 <= 5e+294)
		tmp = (2.0 / t) - 2.0;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+148], t$95$1, If[LessEqual[t$95$3, -1.0], t$95$2, If[LessEqual[t$95$3, 5e+294], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2.0000000000000001e148 or 4.9999999999999999e294 < (/.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 92.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 79.6

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

   5. Applied rewrites 79.6%

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

   if -2.0000000000000001e148 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 66.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 89.1%

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

   if -1 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.9999999999999999e294

   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 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 28.1

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

   5. Applied rewrites 28.1%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. distribute-lft-out N/A

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

   8. Applied rewrites 82.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 54.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -2 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -1:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_3 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1 - 2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
        (t_2 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)))
        (t_3 (+ -2.0 (/ x y))))
   (if (<= t_2 -1e+26)
     t_1
     (if (<= t_2 -2.0) t_3 (if (<= t_2 INFINITY) (- t_1 2.0) t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double t_3 = -2.0 + (x / y);
	double tmp;
	if (t_2 <= -1e+26) {
		tmp = t_1;
	} else if (t_2 <= -2.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1 - 2.0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double t_3 = -2.0 + (x / y);
	double tmp;
	if (t_2 <= -1e+26) {
		tmp = t_1;
	} else if (t_2 <= -2.0) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 - 2.0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((2.0 / z) - -2.0) / t
	t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)
	t_3 = -2.0 + (x / y)
	tmp = 0
	if t_2 <= -1e+26:
		tmp = t_1
	elif t_2 <= -2.0:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1 - 2.0
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z))
	t_3 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (t_2 <= -1e+26)
		tmp = t_1;
	elseif (t_2 <= -2.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(t_1 - 2.0);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((2.0 / z) - -2.0) / t;
	t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	t_3 = -2.0 + (x / y);
	tmp = 0.0;
	if (t_2 <= -1e+26)
		tmp = t_1;
	elseif (t_2 <= -2.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1 - 2.0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+26], t$95$1, If[LessEqual[t$95$2, -2.0], t$95$3, If[LessEqual[t$95$2, Infinity], N[(t$95$1 - 2.0), $MachinePrecision], t$95$3]]]]]]

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)) < -1.00000000000000005e26

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -1.00000000000000005e26 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2 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 57.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 99.1%

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

   if -2 < (/.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 98.5%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate--l+ N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 87.2%

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

4. Final simplification 89.3%

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

Alternative 4: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_2 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{elif}\;t\_1 \leq -2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(-2, t, 2\right), 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)))
        (t_2 (+ -2.0 (/ x y))))
   (if (<= t_1 -1e+26)
     (/ (- (/ 2.0 z) -2.0) t)
     (if (<= t_1 -2.0)
       t_2
       (if (<= t_1 INFINITY) (/ (fma z (fma -2.0 t 2.0) 2.0) (* t z)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double t_2 = -2.0 + (x / y);
	double tmp;
	if (t_1 <= -1e+26) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else if (t_1 <= -2.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(z, fma(-2.0, t, 2.0), 2.0) / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z))
	t_2 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (t_1 <= -1e+26)
		tmp = Float64(Float64(Float64(2.0 / z) - -2.0) / t);
	elseif (t_1 <= -2.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(z, fma(-2.0, t, 2.0), 2.0) / Float64(t * z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+26], N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, -2.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(z * N[(-2.0 * t + 2.0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $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)) < -1.00000000000000005e26

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -1.00000000000000005e26 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2 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 57.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 99.1%

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

   if -2 < (/.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 98.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 57.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 43.8

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

   7. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 43.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. distribute-rgt-in N/A

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

       7. mul-1-neg N/A

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

       8. distribute-lft-neg-in N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. distribute-lft-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lft-identity N/A

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

       15. associate-*r* N/A

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

       16. distribute-rgt-out N/A

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

       17. lower-fma.f64 N/A

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

       18. +-commutative N/A

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

       19. lower-fma.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 85.8

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

   12. Applied rewrites 85.8%

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

4. Final simplification 88.9%

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

Alternative 5: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_3 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
        (t_2 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)))
        (t_3 (+ -2.0 (/ x y))))
   (if (<= t_2 -1e+26)
     t_1
     (if (<= t_2 -1.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double t_3 = -2.0 + (x / y);
	double tmp;
	if (t_2 <= -1e+26) {
		tmp = t_1;
	} else if (t_2 <= -1.0) {
		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 / z) - -2.0) / t;
	double t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double t_3 = -2.0 + (x / y);
	double tmp;
	if (t_2 <= -1e+26) {
		tmp = t_1;
	} else if (t_2 <= -1.0) {
		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 / z) - -2.0) / t
	t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)
	t_3 = -2.0 + (x / y)
	tmp = 0
	if t_2 <= -1e+26:
		tmp = t_1
	elif t_2 <= -1.0:
		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(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z))
	t_3 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (t_2 <= -1e+26)
		tmp = t_1;
	elseif (t_2 <= -1.0)
		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 / z) - -2.0) / t;
	t_2 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	t_3 = -2.0 + (x / y);
	tmp = 0.0;
	if (t_2 <= -1e+26)
		tmp = t_1;
	elseif (t_2 <= -1.0)
		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[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+26], t$95$1, If[LessEqual[t$95$2, -1.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.00000000000000005e26 or -1 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 96.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -1.00000000000000005e26 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 58.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 88.9%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, \frac{2}{z} - -2, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ (fma 2.0 z 2.0) (* t z)) (/ x y))))
   (if (<= (/ x y) -5e+23)
     t_1
     (if (<= (/ x y) 0.001)
       (fma (/ 1.0 t) (- (/ 2.0 z) -2.0) -2.0)
       (if (<= (/ x y) 1e+307) t_1 (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (fma(2.0, z, 2.0) / (t * z)) + (x / y);
	double tmp;
	if ((x / y) <= -5e+23) {
		tmp = t_1;
	} else if ((x / y) <= 0.001) {
		tmp = fma((1.0 / t), ((2.0 / z) - -2.0), -2.0);
	} else if ((x / y) <= 1e+307) {
		tmp = t_1;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(fma(2.0, z, 2.0) / Float64(t * z)) + Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -5e+23)
		tmp = t_1;
	elseif (Float64(x / y) <= 0.001)
		tmp = fma(Float64(1.0 / t), Float64(Float64(2.0 / z) - -2.0), -2.0);
	elseif (Float64(x / y) <= 1e+307)
		tmp = t_1;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+23], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 0.001], N[(N[(1.0 / t), $MachinePrecision] * N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] + -2.0), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1e+307], t$95$1, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -4.9999999999999999e23 or 1e-3 < (/.f64 x y) < 9.99999999999999986e306

   1. Initial program 81.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -4.9999999999999999e23 < (/.f64 x y) < 1e-3

   1. Initial program 82.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. distribute-lft-out N/A

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

   8. Applied rewrites 98.6%

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

   if 9.99999999999999986e306 < (/.f64 x y)

   1. Initial program 61.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 61.5

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, \frac{2}{z} - -2, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, \frac{2}{z} - -2, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ 2.0 (* t z)) (/ x y))))
   (if (<= (/ x y) -4.1e+93)
     t_1
     (if (<= (/ x y) 780000000.0)
       (fma (/ 1.0 t) (- (/ 2.0 z) -2.0) -2.0)
       (if (<= (/ x y) INFINITY) t_1 (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / (t * z)) + (x / y);
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = t_1;
	} else if ((x / y) <= 780000000.0) {
		tmp = fma((1.0 / t), ((2.0 / z) - -2.0), -2.0);
	} else if ((x / y) <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / Float64(t * z)) + Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -4.1e+93)
		tmp = t_1;
	elseif (Float64(x / y) <= 780000000.0)
		tmp = fma(Float64(1.0 / t), Float64(Float64(2.0 / z) - -2.0), -2.0);
	elseif (Float64(x / y) <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -4.1e+93], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 780000000.0], N[(N[(1.0 / t), $MachinePrecision] * N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] + -2.0), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], Infinity], t$95$1, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -4.1000000000000001e93 or 7.8e8 < (/.f64 x y) < +inf.0

   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 z around 0

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

   5. Applied rewrites 87.9%

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

   if -4.1000000000000001e93 < (/.f64 x y) < 7.8e8

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. distribute-lft-out N/A

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

   8. Applied rewrites 95.6%

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

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

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 80.7

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

   7. Applied rewrites 80.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 35.8

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

   10. Applied rewrites 35.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, \frac{2}{z} - -2, -2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{elif}\;\frac{x}{y} \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ 2.0 (* t z)) (/ x y))))
   (if (<= (/ x y) -4.1e+93)
     t_1
     (if (<= (/ x y) 780000000.0)
       (- (/ (- (/ 2.0 z) -2.0) t) 2.0)
       (if (<= (/ x y) INFINITY) t_1 (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / (t * z)) + (x / y);
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = t_1;
	} else if ((x / y) <= 780000000.0) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else if ((x / y) <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / (t * z)) + (x / y);
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = t_1;
	} else if ((x / y) <= 780000000.0) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else if ((x / y) <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 / (t * z)) + (x / y)
	tmp = 0
	if (x / y) <= -4.1e+93:
		tmp = t_1
	elif (x / y) <= 780000000.0:
		tmp = (((2.0 / z) - -2.0) / t) - 2.0
	elif (x / y) <= math.inf:
		tmp = t_1
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / Float64(t * z)) + Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -4.1e+93)
		tmp = t_1;
	elseif (Float64(x / y) <= 780000000.0)
		tmp = Float64(Float64(Float64(Float64(2.0 / z) - -2.0) / t) - 2.0);
	elseif (Float64(x / y) <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 / (t * z)) + (x / y);
	tmp = 0.0;
	if ((x / y) <= -4.1e+93)
		tmp = t_1;
	elseif ((x / y) <= 780000000.0)
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	elseif ((x / y) <= Inf)
		tmp = t_1;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -4.1e+93], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 780000000.0], N[(N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], Infinity], t$95$1, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -4.1000000000000001e93 or 7.8e8 < (/.f64 x y) < +inf.0

   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 z around 0

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

   5. Applied rewrites 87.9%

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

   if -4.1000000000000001e93 < (/.f64 x y) < 7.8e8

   1. Initial program 82.4%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate--l+ N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 95.5%

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

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

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 80.7

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

   7. Applied rewrites 80.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 35.8

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

   10. Applied rewrites 35.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{elif}\;\frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 16500000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -4.1e+93)
   (/ x y)
   (if (<= (/ x y) 16500000.0) (- (/ 2.0 t) 2.0) (+ -2.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 16500000.0) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = -2.0 + (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) <= (-4.1d+93)) then
        tmp = x / y
    else if ((x / y) <= 16500000.0d0) then
        tmp = (2.0d0 / t) - 2.0d0
    else
        tmp = (-2.0d0) + (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) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 16500000.0) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = -2.0 + (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -4.1e+93:
		tmp = x / y
	elif (x / y) <= 16500000.0:
		tmp = (2.0 / t) - 2.0
	else:
		tmp = -2.0 + (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -4.1e+93)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 16500000.0)
		tmp = Float64(Float64(2.0 / t) - 2.0);
	else
		tmp = Float64(-2.0 + Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -4.1e+93)
		tmp = x / y;
	elseif ((x / y) <= 16500000.0)
		tmp = (2.0 / t) - 2.0;
	else
		tmp = -2.0 + (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -4.1e+93], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 16500000.0], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision], N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -4.1000000000000001e93

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 79.6

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 76.2

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

   10. Applied rewrites 76.2%

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

   if -4.1000000000000001e93 < (/.f64 x y) < 1.65e7

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. distribute-lft-out N/A

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 62.3%

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

   if 1.65e7 < (/.f64 x y)

   1. Initial program 78.0%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 71.2%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 16500000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -4.1e+93)
   (/ x y)
   (if (<= (/ x y) 780000000.0) (- (/ 2.0 t) 2.0) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 780000000.0) {
		tmp = (2.0 / t) - 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) <= (-4.1d+93)) then
        tmp = x / y
    else if ((x / y) <= 780000000.0d0) then
        tmp = (2.0d0 / t) - 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) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 780000000.0) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -4.1e+93:
		tmp = x / y
	elif (x / y) <= 780000000.0:
		tmp = (2.0 / t) - 2.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -4.1e+93)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 780000000.0)
		tmp = Float64(Float64(2.0 / t) - 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) <= -4.1e+93)
		tmp = x / y;
	elseif ((x / y) <= 780000000.0)
		tmp = (2.0 / t) - 2.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -4.1e+93], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 780000000.0], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4.1000000000000001e93 or 7.8e8 < (/.f64 x y)

   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. 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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 73.1

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

   10. Applied rewrites 73.1%

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

   if -4.1000000000000001e93 < (/.f64 x y) < 7.8e8

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. distribute-lft-out N/A

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 62.3%

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

Alternative 11: 45.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 780000000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -4.1e+93)
   (/ x y)
   (if (<= (/ x y) 780000000.0) (/ 2.0 t) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 780000000.0) {
		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) <= (-4.1d+93)) then
        tmp = x / y
    else if ((x / y) <= 780000000.0d0) 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) <= -4.1e+93) {
		tmp = x / y;
	} else if ((x / y) <= 780000000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -4.1e+93:
		tmp = x / y
	elif (x / y) <= 780000000.0:
		tmp = 2.0 / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -4.1e+93)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 780000000.0)
		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) <= -4.1e+93)
		tmp = x / y;
	elseif ((x / y) <= 780000000.0)
		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], -4.1e+93], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 780000000.0], N[(2.0 / t), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4.1000000000000001e93 or 7.8e8 < (/.f64 x y)

   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. 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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 73.1

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

   10. Applied rewrites 73.1%

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

   if -4.1000000000000001e93 < (/.f64 x y) < 7.8e8

   1. Initial program 82.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 30.4%

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

Alternative 12: 34.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift-+.f64 N/A

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

   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. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 80.7%

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

5. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 80.7

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

7. Applied rewrites 80.7%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 35.8

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

10. Applied rewrites 35.8%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
11. Add Preprocessing

Developer Target 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024296 
(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))))