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

Percentage Accurate: 86.5% → 99.4%

Time: 10.0s

Alternatives: 11

Speedup: 0.9×

• 23.4% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 10^{+306}:\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 2\right), z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, \frac{2}{z} - -2, x\right)}{y} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)) 1e+306)
   (+ (/ x y) (/ (fma (fma -2.0 t 2.0) z 2.0) (* t z)))
   (- (/ (fma (/ y t) (- (/ 2.0 z) -2.0) x) y) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)) <= 1e+306) {
		tmp = (x / y) + (fma(fma(-2.0, t, 2.0), z, 2.0) / (t * z));
	} else {
		tmp = (fma((y / t), ((2.0 / z) - -2.0), x) / y) - 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z)) <= 1e+306)
		tmp = Float64(Float64(x / y) + Float64(fma(fma(-2.0, t, 2.0), z, 2.0) / Float64(t * z)));
	else
		tmp = Float64(Float64(fma(Float64(y / t), Float64(Float64(2.0 / z) - -2.0), x) / y) - 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(x / y), $MachinePrecision] + N[(N[(N[(-2.0 * t + 2.0), $MachinePrecision] * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / t), $MachinePrecision] * N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.00000000000000002e306

   1. Initial program 99.8%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing
   3. 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. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.00000000000000002e306 < (/.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 29.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. remove-double-neg N/A

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

      7. unsub-neg N/A

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

      8. div-sub N/A

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

      9. distribute-frac-neg N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 10^{+306}:\\ \;\;\;\;\frac{x}{y} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 2\right), z, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, \frac{2}{z} - -2, x\right)}{y} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := -2 + \frac{x}{y}\\ t_3 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z)))
        (t_2 (+ -2.0 (/ x y)))
        (t_3 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z))))
   (if (<= t_3 -5e+43)
     t_1
     (if (<= t_3 2e+46)
       t_2
       (if (<= t_3 4e+202) (/ 2.0 t) (if (<= t_3 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = -2.0 + (x / y);
	double t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double tmp;
	if (t_3 <= -5e+43) {
		tmp = t_1;
	} else if (t_3 <= 2e+46) {
		tmp = t_2;
	} else if (t_3 <= 4e+202) {
		tmp = 2.0 / t;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = -2.0 + (x / y);
	double t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	double tmp;
	if (t_3 <= -5e+43) {
		tmp = t_1;
	} else if (t_3 <= 2e+46) {
		tmp = t_2;
	} else if (t_3 <= 4e+202) {
		tmp = 2.0 / t;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (t * z)
	t_2 = -2.0 + (x / y)
	t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)
	tmp = 0
	if t_3 <= -5e+43:
		tmp = t_1
	elif t_3 <= 2e+46:
		tmp = t_2
	elif t_3 <= 4e+202:
		tmp = 2.0 / t
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	t_2 = Float64(-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 <= -5e+43)
		tmp = t_1;
	elseif (t_3 <= 2e+46)
		tmp = t_2;
	elseif (t_3 <= 4e+202)
		tmp = Float64(2.0 / t);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (t * z);
	t_2 = -2.0 + (x / y);
	t_3 = (((1.0 - t) * (z * 2.0)) + 2.0) / (t * z);
	tmp = 0.0;
	if (t_3 <= -5e+43)
		tmp = t_1;
	elseif (t_3 <= 2e+46)
		tmp = t_2;
	elseif (t_3 <= 4e+202)
		tmp = 2.0 / t;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-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, -5e+43], t$95$1, If[LessEqual[t$95$3, 2e+46], t$95$2, If[LessEqual[t$95$3, 4e+202], N[(2.0 / t), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5.0000000000000004e43 or 3.9999999999999996e202 < (/.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 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 66.2

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 66.2%

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

   if -5.0000000000000004e43 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2e46 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 69.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 92.0%

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

   if 2e46 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 3.9999999999999996e202

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_2 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+43)
     (/ (fma z 2.0 2.0) (* t z))
     (if (<= t_1 50.0)
       t_2
       (if (<= t_1 INFINITY) (/ (- (/ 2.0 z) -2.0) t) 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 <= -5e+43) {
		tmp = fma(z, 2.0, 2.0) / (t * z);
	} else if (t_1 <= 50.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((2.0 / z) - -2.0) / t;
	} 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 <= -5e+43)
		tmp = Float64(fma(z, 2.0, 2.0) / Float64(t * z));
	elseif (t_1 <= 50.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(2.0 / z) - -2.0) / t);
	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, -5e+43], N[(N[(z * 2.0 + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5.0000000000000004e43

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.9%

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

   if -5.0000000000000004e43 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 50 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 66.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.5%

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

   if 50 < (/.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 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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 50:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ t_2 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_3 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma z 2.0 2.0) (* t z)))
        (t_2 (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)))
        (t_3 (+ -2.0 (/ x y))))
   (if (<= t_2 -5e+43)
     t_1
     (if (<= t_2 50.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, 2.0, 2.0) / (t * z);
	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 <= -5e+43) {
		tmp = t_1;
	} else if (t_2 <= 50.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(z, 2.0, 2.0) / Float64(t * z))
	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 <= -5e+43)
		tmp = t_1;
	elseif (t_2 <= 50.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * 2.0 + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $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, -5e+43], t$95$1, If[LessEqual[t$95$2, 50.0], 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)) < -5.0000000000000004e43 or 50 < (/.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 99.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.4%

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

   if -5.0000000000000004e43 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 50 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 66.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 50:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x / y), $MachinePrecision] + N[(N[(N[(N[(1.0 - t), $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x / y), $MachinePrecision] + N[(N[(N[(-2.0 * t + 2.0), $MachinePrecision] * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $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 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 97.4%

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

4. Final simplification 99.5%

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

Alternative 6: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.2:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ 2.0 (* t z)) (/ x y))))
   (if (<= (/ x y) -1e+28)
     t_1
     (if (<= (/ x y) 0.2) (- (/ (- (/ 2.0 z) -2.0) t) 2.0) t_1))))

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) <= -1e+28) {
		tmp = t_1;
	} else if ((x / y) <= 0.2) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / (t * z)) + (x / y)
    if ((x / y) <= (-1d+28)) then
        tmp = t_1
    else if ((x / y) <= 0.2d0) then
        tmp = (((2.0d0 / z) - (-2.0d0)) / t) - 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / (t * z)) + (x / y);
	double tmp;
	if ((x / y) <= -1e+28) {
		tmp = t_1;
	} else if ((x / y) <= 0.2) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 / (t * z)) + (x / y)
	tmp = 0
	if (x / y) <= -1e+28:
		tmp = t_1
	elif (x / y) <= 0.2:
		tmp = (((2.0 / z) - -2.0) / t) - 2.0
	else:
		tmp = t_1
	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) <= -1e+28)
		tmp = t_1;
	elseif (Float64(x / y) <= 0.2)
		tmp = Float64(Float64(Float64(Float64(2.0 / z) - -2.0) / t) - 2.0);
	else
		tmp = t_1;
	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) <= -1e+28)
		tmp = t_1;
	elseif ((x / y) <= 0.2)
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	else
		tmp = t_1;
	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], -1e+28], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 0.2], N[(N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -9.99999999999999958e27 or 0.20000000000000001 < (/.f64 x y)

   1. Initial program 82.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 92.4%

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

   if -9.99999999999999958e27 < (/.f64 x y) < 0.20000000000000001

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   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 98.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.2:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot z} + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t} + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ 2.0 t) (/ x y))))
   (if (<= (/ x y) -1e+53)
     t_1
     (if (<= (/ x y) 8e+36) (- (/ (- (/ 2.0 z) -2.0) t) 2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / t) + (x / y);
	double tmp;
	if ((x / y) <= -1e+53) {
		tmp = t_1;
	} else if ((x / y) <= 8e+36) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / t) + (x / y)
    if ((x / y) <= (-1d+53)) then
        tmp = t_1
    else if ((x / y) <= 8d+36) then
        tmp = (((2.0d0 / z) - (-2.0d0)) / t) - 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / t) + (x / y);
	double tmp;
	if ((x / y) <= -1e+53) {
		tmp = t_1;
	} else if ((x / y) <= 8e+36) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 / t) + (x / y)
	tmp = 0
	if (x / y) <= -1e+53:
		tmp = t_1
	elif (x / y) <= 8e+36:
		tmp = (((2.0 / z) - -2.0) / t) - 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / t) + Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -1e+53)
		tmp = t_1;
	elseif (Float64(x / y) <= 8e+36)
		tmp = Float64(Float64(Float64(Float64(2.0 / z) - -2.0) / t) - 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 / t) + (x / y);
	tmp = 0.0;
	if ((x / y) <= -1e+53)
		tmp = t_1;
	elseif ((x / y) <= 8e+36)
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e+53], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 8e+36], N[(N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -9.9999999999999999e52 or 8.00000000000000034e36 < (/.f64 x y)

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.3%

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

   if -9.9999999999999999e52 < (/.f64 x y) < 8.00000000000000034e36

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   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 96.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{t} + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{2}{t} - 2\right) + \frac{x}{y}\\ \mathbf{if}\;z \leq -80000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot t, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- (/ 2.0 t) 2.0) (/ x y))))
   (if (<= z -80000000.0)
     t_1
     (if (<= z 7e-6) (+ (/ (fma (* -2.0 t) z 2.0) (* t z)) (/ x y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / t) - 2.0) + (x / y);
	double tmp;
	if (z <= -80000000.0) {
		tmp = t_1;
	} else if (z <= 7e-6) {
		tmp = (fma((-2.0 * t), z, 2.0) / (t * z)) + (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / t) - 2.0) + Float64(x / y))
	tmp = 0.0
	if (z <= -80000000.0)
		tmp = t_1;
	elseif (z <= 7e-6)
		tmp = Float64(Float64(fma(Float64(-2.0 * t), z, 2.0) / Float64(t * z)) + Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -80000000.0], t$95$1, If[LessEqual[z, 7e-6], N[(N[(N[(N[(-2.0 * t), $MachinePrecision] * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e7 or 6.99999999999999989e-6 < z

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -8e7 < z < 6.99999999999999989e-6

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      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. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -80000000:\\ \;\;\;\;\left(\frac{2}{t} - 2\right) + \frac{x}{y}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot t, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} - 2\right) + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -9.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 20000000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -9.6e+20)
   (/ x y)
   (if (<= (/ x y) 20000000.0) (/ 2.0 t) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -9.6e+20) {
		tmp = x / y;
	} else if ((x / y) <= 20000000.0) {
		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) <= (-9.6d+20)) then
        tmp = x / y
    else if ((x / y) <= 20000000.0d0) 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) <= -9.6e+20) {
		tmp = x / y;
	} else if ((x / y) <= 20000000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -9.6e+20:
		tmp = x / y
	elif (x / y) <= 20000000.0:
		tmp = 2.0 / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -9.6e+20)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 20000000.0)
		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) <= -9.6e+20)
		tmp = x / y;
	elseif ((x / y) <= 20000000.0)
		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], -9.6e+20], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 20000000.0], N[(2.0 / t), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -9.6e20 or 2e7 < (/.f64 x y)

   1. Initial program 82.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -9.6e20 < (/.f64 x y) < 2e7

   1. Initial program 86.4%

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

   3. Taylor expanded in t around 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 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 26.8%

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

Alternative 10: 60.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 + \frac{x}{y}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ -2.0 (/ x y))))
   (if (<= t -4.5e-164) t_1 (if (<= t 4.4e-47) (/ 2.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -2.0 + (x / y);
	double tmp;
	if (t <= -4.5e-164) {
		tmp = t_1;
	} else if (t <= 4.4e-47) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-2.0d0) + (x / y)
    if (t <= (-4.5d-164)) then
        tmp = t_1
    else if (t <= 4.4d-47) then
        tmp = 2.0d0 / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -2.0 + (x / y);
	double tmp;
	if (t <= -4.5e-164) {
		tmp = t_1;
	} else if (t <= 4.4e-47) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -2.0 + (x / y)
	tmp = 0
	if t <= -4.5e-164:
		tmp = t_1
	elif t <= 4.4e-47:
		tmp = 2.0 / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (t <= -4.5e-164)
		tmp = t_1;
	elseif (t <= 4.4e-47)
		tmp = Float64(2.0 / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -2.0 + (x / y);
	tmp = 0.0;
	if (t <= -4.5e-164)
		tmp = t_1;
	elseif (t <= 4.4e-47)
		tmp = 2.0 / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e-164], t$95$1, If[LessEqual[t, 4.4e-47], N[(2.0 / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.4999999999999997e-164 or 4.40000000000000037e-47 < t

   1. Initial program 79.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 70.2%

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

   if -4.4999999999999997e-164 < t < 4.40000000000000037e-47

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   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 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% 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 84.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 36.5

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

5. Applied rewrites 36.5%

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

Developer Target 1: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024244 
(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))))