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

Percentage Accurate: 86.3% → 99.3%

Time: 9.1s

Alternatives: 12

Speedup: 0.7×

• 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: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(-2.0 - N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] + 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 99.7%

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

   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 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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 99.0%

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

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ t_2 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} + \frac{x}{y}\\ t_3 := \left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ (fma 2.0 z 2.0) (* t z)) (/ x y)))
        (t_2 (+ (/ (+ (* (- 1.0 t) (* z 2.0)) 2.0) (* t z)) (/ x y)))
        (t_3 (+ (- -2.0 (/ -2.0 t)) (/ x y))))
   (if (<= t_2 -5e+36)
     t_1
     (if (<= t_2 50000000000.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (fma(2.0, z, 2.0) / (t * z)) + (x / y);
	double t_2 = ((((1.0 - t) * (z * 2.0)) + 2.0) / (t * z)) + (x / y);
	double t_3 = (-2.0 - (-2.0 / t)) + (x / y);
	double tmp;
	if (t_2 <= -5e+36) {
		tmp = t_1;
	} else if (t_2 <= 50000000000.0) {
		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(Float64(fma(2.0, z, 2.0) / Float64(t * z)) + Float64(x / y))
	t_2 = Float64(Float64(Float64(Float64(Float64(1.0 - t) * Float64(z * 2.0)) + 2.0) / Float64(t * z)) + Float64(x / y))
	t_3 = Float64(Float64(-2.0 - Float64(-2.0 / t)) + Float64(x / y))
	tmp = 0.0
	if (t_2 <= -5e+36)
		tmp = t_1;
	elseif (t_2 <= 50000000000.0)
		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[(N[(2.0 * z + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(-2.0 - N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+36], t$95$1, If[LessEqual[t$95$2, 50000000000.0], 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 x y) (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))) < -4.99999999999999977e36 or 5e10 < (+.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 99.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2 + 2 \cdot z}}{t \cdot z} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   if -4.99999999999999977e36 < (+.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))) < 5e10 or +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 50.5%

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

   3. Taylor expanded in z around 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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} + \frac{x}{y} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} + \frac{x}{y} \leq 50000000000:\\ \;\;\;\;\left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} + \frac{x}{y} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{t \cdot z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ t_2 := \frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z}\\ t_3 := -2 + \frac{x}{y}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -1e+27)
     t_1
     (if (<= t_2 10000.0) 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 <= -1e+27) {
		tmp = t_1;
	} else if (t_2 <= 10000.0) {
		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 <= -1e+27)
		tmp = t_1;
	elseif (t_2 <= 10000.0)
		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, -1e+27], t$95$1, If[LessEqual[t$95$2, 10000.0], 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)) < -1e27 or 1e4 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 97.0%

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

   3. Taylor expanded in t around 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 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 78.3%

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

   if -1e27 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1e4 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 65.7%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 96.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq 10000:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{\left(1 - t\right) \cdot \left(z \cdot 2\right) + 2}{t \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -1.18 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 1.18 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- -2.0 (/ -2.0 t)) (/ x y))))
   (if (<= (/ x y) -1.18e+70)
     t_1
     (if (<= (/ x y) 1.18e+47) (- (/ (- (/ 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 - (-2.0 / t)) + (x / y);
	double tmp;
	if ((x / y) <= -1.18e+70) {
		tmp = t_1;
	} else if ((x / y) <= 1.18e+47) {
		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) - ((-2.0d0) / t)) + (x / y)
    if ((x / y) <= (-1.18d+70)) then
        tmp = t_1
    else if ((x / y) <= 1.18d+47) 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 - (-2.0 / t)) + (x / y);
	double tmp;
	if ((x / y) <= -1.18e+70) {
		tmp = t_1;
	} else if ((x / y) <= 1.18e+47) {
		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 - (-2.0 / t)) + (x / y)
	tmp = 0
	if (x / y) <= -1.18e+70:
		tmp = t_1
	elif (x / y) <= 1.18e+47:
		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(-2.0 / t)) + Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -1.18e+70)
		tmp = t_1;
	elseif (Float64(x / y) <= 1.18e+47)
		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 - (-2.0 / t)) + (x / y);
	tmp = 0.0;
	if ((x / y) <= -1.18e+70)
		tmp = t_1;
	elseif ((x / y) <= 1.18e+47)
		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[(-2.0 / t), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1.18e+70], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 1.18e+47], 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) < -1.18000000000000001e70 or 1.18e47 < (/.f64 x y)

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.18000000000000001e70 < (/.f64 x y) < 1.18e47

   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 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 95.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.18 \cdot 10^{+70}:\\ \;\;\;\;\left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.18 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 - \frac{-2}{t}\right) + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5.5e+78)
   (/ x y)
   (if (<= (/ x y) 2.6e+95) (- (/ (- (/ 2.0 z) -2.0) t) 2.0) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -5.5e+78) {
		tmp = x / y;
	} else if ((x / y) <= 2.6e+95) {
		tmp = (((2.0 / z) - -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) <= (-5.5d+78)) then
        tmp = x / y
    else if ((x / y) <= 2.6d+95) then
        tmp = (((2.0d0 / z) - (-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) <= -5.5e+78) {
		tmp = x / y;
	} else if ((x / y) <= 2.6e+95) {
		tmp = (((2.0 / z) - -2.0) / t) - 2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -5.5e+78:
		tmp = x / y
	elif (x / y) <= 2.6e+95:
		tmp = (((2.0 / z) - -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) <= -5.5e+78)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2.6e+95)
		tmp = Float64(Float64(Float64(Float64(2.0 / z) - -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) <= -5.5e+78)
		tmp = x / y;
	elseif ((x / y) <= 2.6e+95)
		tmp = (((2.0 / z) - -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], -5.5e+78], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2.6e+95], N[(N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5.4999999999999997e78 or 2.5999999999999999e95 < (/.f64 x y)

   1. Initial program 79.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. 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 80.0%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 79.5

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

   10. Applied rewrites 79.5%

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

   if -5.4999999999999997e78 < (/.f64 x y) < 2.5999999999999999e95

   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 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 93.8%

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

Alternative 6: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5.5e+78)
   (/ x y)
   (if (<= (/ x y) 2.6e+95) (- (/ (fma z 2.0 2.0) (* t z)) 2.0) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -5.5e+78) {
		tmp = x / y;
	} else if ((x / y) <= 2.6e+95) {
		tmp = (fma(z, 2.0, 2.0) / (t * z)) - 2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -5.5e+78)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2.6e+95)
		tmp = Float64(Float64(fma(z, 2.0, 2.0) / Float64(t * z)) - 2.0);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -5.5e+78], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2.6e+95], N[(N[(N[(z * 2.0 + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5.4999999999999997e78 or 2.5999999999999999e95 < (/.f64 x y)

   1. Initial program 79.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. 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 80.0%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 79.5

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

   10. Applied rewrites 79.5%

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

   if -5.4999999999999997e78 < (/.f64 x y) < 2.5999999999999999e95

   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 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 93.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.6%

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

Alternative 7: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.75 \cdot 10^{-12}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1.75e-12)
   (+ -2.0 (/ x y))
   (if (<= (/ x y) 7.8e+30) (- (/ 2.0 t) 2.0) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1.75e-12) {
		tmp = -2.0 + (x / y);
	} else if ((x / y) <= 7.8e+30) {
		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) <= (-1.75d-12)) then
        tmp = (-2.0d0) + (x / y)
    else if ((x / y) <= 7.8d+30) 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) <= -1.75e-12) {
		tmp = -2.0 + (x / y);
	} else if ((x / y) <= 7.8e+30) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -1.75e-12:
		tmp = -2.0 + (x / y)
	elif (x / y) <= 7.8e+30:
		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) <= -1.75e-12)
		tmp = Float64(-2.0 + Float64(x / y));
	elseif (Float64(x / y) <= 7.8e+30)
		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) <= -1.75e-12)
		tmp = -2.0 + (x / y);
	elseif ((x / y) <= 7.8e+30)
		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], -1.75e-12], N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 7.8e+30], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.75e-12

   1. Initial program 81.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 71.7%

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

   if -1.75e-12 < (/.f64 x y) < 7.80000000000000021e30

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   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 99.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.0%

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

   if 7.80000000000000021e30 < (/.f64 x y)

   1. Initial program 79.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 79.5%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 66.5

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

   10. Applied rewrites 66.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.75 \cdot 10^{-12}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1.5e+38)
   (/ x y)
   (if (<= (/ x y) 7.8e+30) (- (/ 2.0 t) 2.0) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 7.8e+30) {
		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) <= (-1.5d+38)) then
        tmp = x / y
    else if ((x / y) <= 7.8d+30) 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) <= -1.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 7.8e+30) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -1.5e+38:
		tmp = x / y
	elif (x / y) <= 7.8e+30:
		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) <= -1.5e+38)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 7.8e+30)
		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) <= -1.5e+38)
		tmp = x / y;
	elseif ((x / y) <= 7.8e+30)
		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], -1.5e+38], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 7.8e+30], 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) < -1.5000000000000001e38 or 7.80000000000000021e30 < (/.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. 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 82.6%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 71.8

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

   10. Applied rewrites 71.8%

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

   if -1.5000000000000001e38 < (/.f64 x y) < 7.80000000000000021e30

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 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 97.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 63.4%

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

Alternative 9: 46.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1.5e+38) (/ x y) (if (<= (/ x y) 8e+16) (/ 2.0 t) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 8e+16) {
		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) <= (-1.5d+38)) then
        tmp = x / y
    else if ((x / y) <= 8d+16) 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) <= -1.5e+38) {
		tmp = x / y;
	} else if ((x / y) <= 8e+16) {
		tmp = 2.0 / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -1.5e+38:
		tmp = x / y
	elif (x / y) <= 8e+16:
		tmp = 2.0 / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1.5e+38)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 8e+16)
		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) <= -1.5e+38)
		tmp = x / y;
	elseif ((x / y) <= 8e+16)
		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], -1.5e+38], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 8e+16], N[(2.0 / t), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.5000000000000001e38 or 8e16 < (/.f64 x y)

   1. Initial program 82.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. 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 82.9%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if -1.5000000000000001e38 < (/.f64 x y) < 8e16

   1. Initial program 85.1%

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

   3. Taylor expanded in 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 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 28.4%

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

Alternative 10: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t} - 2\\ t_2 := -2 + \frac{x}{y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+232}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-79}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{t \cdot z} - 2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+176}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ 2.0 t) 2.0)) (t_2 (+ -2.0 (/ x y))))
   (if (<= z -9.2e+232)
     t_2
     (if (<= z -3.2e+76)
       t_1
       (if (<= z -1.35e-79)
         t_2
         (if (<= z 7.2e-40)
           (- (/ 2.0 (* t z)) 2.0)
           (if (<= z 2.6e+176) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / t) - 2.0;
	double t_2 = -2.0 + (x / y);
	double tmp;
	if (z <= -9.2e+232) {
		tmp = t_2;
	} else if (z <= -3.2e+76) {
		tmp = t_1;
	} else if (z <= -1.35e-79) {
		tmp = t_2;
	} else if (z <= 7.2e-40) {
		tmp = (2.0 / (t * z)) - 2.0;
	} else if (z <= 2.6e+176) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 / t) - 2.0d0
    t_2 = (-2.0d0) + (x / y)
    if (z <= (-9.2d+232)) then
        tmp = t_2
    else if (z <= (-3.2d+76)) then
        tmp = t_1
    else if (z <= (-1.35d-79)) then
        tmp = t_2
    else if (z <= 7.2d-40) then
        tmp = (2.0d0 / (t * z)) - 2.0d0
    else if (z <= 2.6d+176) then
        tmp = t_2
    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) - 2.0;
	double t_2 = -2.0 + (x / y);
	double tmp;
	if (z <= -9.2e+232) {
		tmp = t_2;
	} else if (z <= -3.2e+76) {
		tmp = t_1;
	} else if (z <= -1.35e-79) {
		tmp = t_2;
	} else if (z <= 7.2e-40) {
		tmp = (2.0 / (t * z)) - 2.0;
	} else if (z <= 2.6e+176) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 / t) - 2.0
	t_2 = -2.0 + (x / y)
	tmp = 0
	if z <= -9.2e+232:
		tmp = t_2
	elif z <= -3.2e+76:
		tmp = t_1
	elif z <= -1.35e-79:
		tmp = t_2
	elif z <= 7.2e-40:
		tmp = (2.0 / (t * z)) - 2.0
	elif z <= 2.6e+176:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / t) - 2.0)
	t_2 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (z <= -9.2e+232)
		tmp = t_2;
	elseif (z <= -3.2e+76)
		tmp = t_1;
	elseif (z <= -1.35e-79)
		tmp = t_2;
	elseif (z <= 7.2e-40)
		tmp = Float64(Float64(2.0 / Float64(t * z)) - 2.0);
	elseif (z <= 2.6e+176)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 / t) - 2.0;
	t_2 = -2.0 + (x / y);
	tmp = 0.0;
	if (z <= -9.2e+232)
		tmp = t_2;
	elseif (z <= -3.2e+76)
		tmp = t_1;
	elseif (z <= -1.35e-79)
		tmp = t_2;
	elseif (z <= 7.2e-40)
		tmp = (2.0 / (t * z)) - 2.0;
	elseif (z <= 2.6e+176)
		tmp = t_2;
	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] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+232], t$95$2, If[LessEqual[z, -3.2e+76], t$95$1, If[LessEqual[z, -1.35e-79], t$95$2, If[LessEqual[z, 7.2e-40], N[(N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], If[LessEqual[z, 2.6e+176], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.20000000000000024e232 or -3.19999999999999976e76 < z < -1.3500000000000001e-79 or 7.2e-40 < z < 2.59999999999999991e176

   1. Initial program 75.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 79.7%

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

   if -9.20000000000000024e232 < z < -3.19999999999999976e76 or 2.59999999999999991e176 < z

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   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 76.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.6%

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

   if -1.3500000000000001e-79 < z < 7.2e-40

   1. Initial program 95.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 81.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 81.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+232}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-79}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{t \cdot z} - 2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+176}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t} - 2\\ t_2 := -2 + \frac{x}{y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+232}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+176}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ 2.0 t) 2.0)) (t_2 (+ -2.0 (/ x y))))
   (if (<= z -9.2e+232)
     t_2
     (if (<= z -3.2e+76)
       t_1
       (if (<= z -6.5e-81)
         t_2
         (if (<= z 6.8e-40) (/ 2.0 (* t z)) (if (<= z 2.6e+176) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / t) - 2.0;
	double t_2 = -2.0 + (x / y);
	double tmp;
	if (z <= -9.2e+232) {
		tmp = t_2;
	} else if (z <= -3.2e+76) {
		tmp = t_1;
	} else if (z <= -6.5e-81) {
		tmp = t_2;
	} else if (z <= 6.8e-40) {
		tmp = 2.0 / (t * z);
	} else if (z <= 2.6e+176) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 / t) - 2.0d0
    t_2 = (-2.0d0) + (x / y)
    if (z <= (-9.2d+232)) then
        tmp = t_2
    else if (z <= (-3.2d+76)) then
        tmp = t_1
    else if (z <= (-6.5d-81)) then
        tmp = t_2
    else if (z <= 6.8d-40) then
        tmp = 2.0d0 / (t * z)
    else if (z <= 2.6d+176) then
        tmp = t_2
    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) - 2.0;
	double t_2 = -2.0 + (x / y);
	double tmp;
	if (z <= -9.2e+232) {
		tmp = t_2;
	} else if (z <= -3.2e+76) {
		tmp = t_1;
	} else if (z <= -6.5e-81) {
		tmp = t_2;
	} else if (z <= 6.8e-40) {
		tmp = 2.0 / (t * z);
	} else if (z <= 2.6e+176) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 / t) - 2.0
	t_2 = -2.0 + (x / y)
	tmp = 0
	if z <= -9.2e+232:
		tmp = t_2
	elif z <= -3.2e+76:
		tmp = t_1
	elif z <= -6.5e-81:
		tmp = t_2
	elif z <= 6.8e-40:
		tmp = 2.0 / (t * z)
	elif z <= 2.6e+176:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / t) - 2.0)
	t_2 = Float64(-2.0 + Float64(x / y))
	tmp = 0.0
	if (z <= -9.2e+232)
		tmp = t_2;
	elseif (z <= -3.2e+76)
		tmp = t_1;
	elseif (z <= -6.5e-81)
		tmp = t_2;
	elseif (z <= 6.8e-40)
		tmp = Float64(2.0 / Float64(t * z));
	elseif (z <= 2.6e+176)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 / t) - 2.0;
	t_2 = -2.0 + (x / y);
	tmp = 0.0;
	if (z <= -9.2e+232)
		tmp = t_2;
	elseif (z <= -3.2e+76)
		tmp = t_1;
	elseif (z <= -6.5e-81)
		tmp = t_2;
	elseif (z <= 6.8e-40)
		tmp = 2.0 / (t * z);
	elseif (z <= 2.6e+176)
		tmp = t_2;
	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] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+232], t$95$2, If[LessEqual[z, -3.2e+76], t$95$1, If[LessEqual[z, -6.5e-81], t$95$2, If[LessEqual[z, 6.8e-40], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e+176], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.20000000000000024e232 or -3.19999999999999976e76 < z < -6.5000000000000002e-81 or 6.79999999999999968e-40 < z < 2.59999999999999991e176

   1. Initial program 75.4%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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 79.7%

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

   if -9.20000000000000024e232 < z < -3.19999999999999976e76 or 2.59999999999999991e176 < z

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   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 76.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.6%

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

   if -6.5000000000000002e-81 < z < 6.79999999999999968e-40

   1. Initial program 95.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+232}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+176}:\\ \;\;\;\;-2 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. 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 84.2%

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

   1. lift-+.f64 N/A

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

   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 84.1%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 76.7

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

7. Applied rewrites 76.7%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 32.7

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

10. Applied rewrites 32.7%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
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 2024294 
(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))))