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

Percentage Accurate: 86.1% → 99.4%

Time: 11.9s

Alternatives: 12

Speedup: 0.8×

• 23.6% 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: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x / y), $MachinePrecision] + N[(N[(z * N[(-2.0 * t + 2.0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $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 99.3%

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.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. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 N/A

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

      19. /-lowering-/.f64 99.3

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

   4. Applied egg-rr 99.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.4%

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

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (/ 2.0 z)) t))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t)))
        (t_3 (+ (/ x y) -2.0)))
   (if (<= t_2 -2e+38)
     t_1
     (if (<= t_2 1000000000000.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -2e+38) {
		tmp = t_1;
	} else if (t_2 <= 1000000000000.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + (2.0 / z)) / t;
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -2e+38) {
		tmp = t_1;
	} else if (t_2 <= 1000000000000.0) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + (2.0 / z)) / t
	t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t)
	t_3 = (x / y) + -2.0
	tmp = 0
	if t_2 <= -2e+38:
		tmp = t_1
	elif t_2 <= 1000000000000.0:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(2.0 / z)) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -2e+38)
		tmp = t_1;
	elseif (t_2 <= 1000000000000.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + (2.0 / z)) / t;
	t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	t_3 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_2 <= -2e+38)
		tmp = t_1;
	elseif (t_2 <= 1000000000000.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+38], t$95$1, If[LessEqual[t$95$2, 1000000000000.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.99999999999999995e38 or 1e12 < (/.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 97.9%

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

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 80.2

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

   8. Simplified 80.2%

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

   if -1.99999999999999995e38 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1e12 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 59.3%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 94.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 1000000000000:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ t_2 := \frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 z 2.0) (* z t)))
        (t_2 (/ (+ 2.0 (* (* 2.0 z) (- 1.0 t))) (* z t)))
        (t_3 (+ (/ x y) -2.0)))
   (if (<= t_2 -2e+38)
     t_1
     (if (<= t_2 1000000000000.0) 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) / (z * t);
	double t_2 = (2.0 + ((2.0 * z) * (1.0 - t))) / (z * t);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -2e+38) {
		tmp = t_1;
	} else if (t_2 <= 1000000000000.0) {
		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(2.0, z, 2.0) / Float64(z * t))
	t_2 = Float64(Float64(2.0 + Float64(Float64(2.0 * z) * Float64(1.0 - t))) / Float64(z * t))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -2e+38)
		tmp = t_1;
	elseif (t_2 <= 1000000000000.0)
		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[(2.0 * z + 2.0), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+38], t$95$1, If[LessEqual[t$95$2, 1000000000000.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.99999999999999995e38 or 1e12 < (/.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 97.9%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 80.1%

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

   if -1.99999999999999995e38 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1e12 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 59.3%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 94.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq 1000000000000:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{2 + \left(2 \cdot z\right) \cdot \left(1 - t\right)}{z \cdot t} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, z, 2\right)}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, -2, 2 + \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 (* z t)))))
   (if (<= (/ x y) -1e+50)
     t_1
     (if (<= (/ x y) 10000000000000.0)
       (/ (fma t -2.0 (+ 2.0 (/ 2.0 z))) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / (z * t));
	double tmp;
	if ((x / y) <= -1e+50) {
		tmp = t_1;
	} else if ((x / y) <= 10000000000000.0) {
		tmp = fma(t, -2.0, (2.0 + (2.0 / z))) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / Float64(z * t)))
	tmp = 0.0
	if (Float64(x / y) <= -1e+50)
		tmp = t_1;
	elseif (Float64(x / y) <= 10000000000000.0)
		tmp = Float64(fma(t, -2.0, Float64(2.0 + Float64(2.0 / z))) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e+50], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 10000000000000.0], N[(N[(t * -2.0 + N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.0000000000000001e50 or 1e13 < (/.f64 x y)

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 89.9

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

   5. Simplified 89.9%

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

   if -1.0000000000000001e50 < (/.f64 x y) < 1e13

   1. Initial program 85.6%

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

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 98.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, -2, 2 + \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + t\_1\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{x}{y} \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* z t))) (t_2 (+ (/ x y) t_1)))
   (if (<= (/ x y) -1e+50)
     t_2
     (if (<= (/ x y) 10000000000000.0) (fma t_1 (+ z 1.0) -2.0) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (z * t);
	double t_2 = (x / y) + t_1;
	double tmp;
	if ((x / y) <= -1e+50) {
		tmp = t_2;
	} else if ((x / y) <= 10000000000000.0) {
		tmp = fma(t_1, (z + 1.0), -2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(z * t))
	t_2 = Float64(Float64(x / y) + t_1)
	tmp = 0.0
	if (Float64(x / y) <= -1e+50)
		tmp = t_2;
	elseif (Float64(x / y) <= 10000000000000.0)
		tmp = fma(t_1, Float64(z + 1.0), -2.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e+50], t$95$2, If[LessEqual[N[(x / y), $MachinePrecision], 10000000000000.0], N[(t$95$1 * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1.0000000000000001e50 or 1e13 < (/.f64 x y)

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 89.9

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

   5. Simplified 89.9%

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

   if -1.0000000000000001e50 < (/.f64 x y) < 1e13

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e+61)
   (+ (/ x y) (/ 2.0 t))
   (if (<= (/ x y) 1e+97)
     (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0)
     (+ (/ x y) (+ -2.0 (/ 2.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e+61) {
		tmp = (x / y) + (2.0 / t);
	} else if ((x / y) <= 1e+97) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = (x / y) + (-2.0 + (2.0 / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1e+61)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (Float64(x / y) <= 1e+97)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = Float64(Float64(x / y) + Float64(-2.0 + Float64(2.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -1e+61], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1e+97], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -9.99999999999999949e60

   1. Initial program 81.5%

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

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 78.5

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

   5. Simplified 78.5%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 78.5

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

   8. Simplified 78.5%

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

   if -9.99999999999999949e60 < (/.f64 x y) < 1.0000000000000001e97

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 95.7%

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

   if 1.0000000000000001e97 < (/.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. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 82.3

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

   5. Simplified 82.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 t))))
   (if (<= (/ x y) -1e+61)
     t_1
     (if (<= (/ x y) 1e+97) (fma (/ 2.0 (* z t)) (+ z 1.0) -2.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / t);
	double tmp;
	if ((x / y) <= -1e+61) {
		tmp = t_1;
	} else if ((x / y) <= 1e+97) {
		tmp = fma((2.0 / (z * t)), (z + 1.0), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / t))
	tmp = 0.0
	if (Float64(x / y) <= -1e+61)
		tmp = t_1;
	elseif (Float64(x / y) <= 1e+97)
		tmp = fma(Float64(2.0 / Float64(z * t)), Float64(z + 1.0), -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e+61], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 1e+97], N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(z + 1.0), $MachinePrecision] + -2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -9.99999999999999949e60 or 1.0000000000000001e97 < (/.f64 x y)

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 80.3

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

   5. Simplified 80.3%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 80.3

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

   8. Simplified 80.3%

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

   if -9.99999999999999949e60 < (/.f64 x y) < 1.0000000000000001e97

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 95.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{z \cdot t}, z + 1, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -7.4e+49)
   (/ x y)
   (if (<= (/ x y) 1.5e+23) (+ -2.0 (/ 2.0 t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -7.4e+49) {
		tmp = x / y;
	} else if ((x / y) <= 1.5e+23) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x / y) <= (-7.4d+49)) then
        tmp = x / y
    else if ((x / y) <= 1.5d+23) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -7.4e+49) {
		tmp = x / y;
	} else if ((x / y) <= 1.5e+23) {
		tmp = -2.0 + (2.0 / t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -7.4e+49:
		tmp = x / y
	elif (x / y) <= 1.5e+23:
		tmp = -2.0 + (2.0 / t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -7.4e+49)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 1.5e+23)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -7.4e+49)
		tmp = x / y;
	elseif ((x / y) <= 1.5e+23)
		tmp = -2.0 + (2.0 / t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -7.4e+49], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1.5e+23], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -7.40000000000000036e49 or 1.5e23 < (/.f64 x y)

   1. Initial program 82.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 67.5

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

   5. Simplified 67.5%

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

   if -7.40000000000000036e49 < (/.f64 x y) < 1.5e23

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.2%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 65.9

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

   8. Simplified 65.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.75 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -6.8e+49)
   (/ x y)
   (if (<= (/ x y) 2.75e+19) (/ 2.0 t) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -6.8e+49) {
		tmp = x / y;
	} else if ((x / y) <= 2.75e+19) {
		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) <= (-6.8d+49)) then
        tmp = x / y
    else if ((x / y) <= 2.75d+19) 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) <= -6.8e+49) {
		tmp = x / y;
	} else if ((x / y) <= 2.75e+19) {
		tmp = 2.0 / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -6.8e+49:
		tmp = x / y
	elif (x / y) <= 2.75e+19:
		tmp = 2.0 / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -6.8e+49)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2.75e+19)
		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) <= -6.8e+49)
		tmp = x / y;
	elseif ((x / y) <= 2.75e+19)
		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], -6.8e+49], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2.75e+19], N[(2.0 / t), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -6.8000000000000001e49 or 2.75e19 < (/.f64 x y)

   1. Initial program 82.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 67.5

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

   5. Simplified 67.5%

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

   if -6.8000000000000001e49 < (/.f64 x y) < 2.75e19

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 67.3

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

   5. Simplified 67.3%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 39.1

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

   8. Simplified 39.1%

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

Alternative 10: 62.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.05:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq 10^{-88}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.05)
   (+ -2.0 (/ 2.0 t))
   (if (<= z 1e-88) (/ 2.0 (* z t)) (+ (/ x y) -2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.05) {
		tmp = -2.0 + (2.0 / t);
	} else if (z <= 1e-88) {
		tmp = 2.0 / (z * t);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-0.05d0)) then
        tmp = (-2.0d0) + (2.0d0 / t)
    else if (z <= 1d-88) then
        tmp = 2.0d0 / (z * t)
    else
        tmp = (x / y) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.05) {
		tmp = -2.0 + (2.0 / t);
	} else if (z <= 1e-88) {
		tmp = 2.0 / (z * t);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.05:
		tmp = -2.0 + (2.0 / t)
	elif z <= 1e-88:
		tmp = 2.0 / (z * t)
	else:
		tmp = (x / y) + -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.05)
		tmp = Float64(-2.0 + Float64(2.0 / t));
	elseif (z <= 1e-88)
		tmp = Float64(2.0 / Float64(z * t));
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.05)
		tmp = -2.0 + (2.0 / t);
	elseif (z <= 1e-88)
		tmp = 2.0 / (z * t);
	else
		tmp = (x / y) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.05], N[(-2.0 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-88], N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.050000000000000003

   1. Initial program 74.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.9%

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

   6. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 72.3

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

   8. Simplified 72.3%

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

   if -0.050000000000000003 < z < 9.99999999999999934e-89

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 69.4

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

   5. Simplified 69.4%

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

   if 9.99999999999999934e-89 < z

   1. Initial program 77.4%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 60.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.05:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;z \leq 10^{-88}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-24}:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.4e-24) -2.0 (if (<= t 1.0) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.4e-24) {
		tmp = -2.0;
	} else if (t <= 1.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.4d-24)) then
        tmp = -2.0d0
    else if (t <= 1.0d0) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.4e-24) {
		tmp = -2.0;
	} else if (t <= 1.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.4e-24:
		tmp = -2.0
	elif t <= 1.0:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.4e-24)
		tmp = -2.0;
	elseif (t <= 1.0)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.4e-24)
		tmp = -2.0;
	elseif (t <= 1.0)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.4e-24], -2.0, If[LessEqual[t, 1.0], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -4.40000000000000003e-24 or 1 < t

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 57.1%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 36.1%

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

   if -4.40000000000000003e-24 < t < 1

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 65.8

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

   5. Simplified 65.8%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 44.5

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

   8. Simplified 44.5%

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

Alternative 12: 20.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 69.2%

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

6. Taylor expanded in t around inf

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

8. Simplified 17.3%

   \[\leadsto \color{blue}{-2} \]
9. 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 2024204 
(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))))