Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 7

Speedup: 1.0×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (* (+ (* x y) z) y) t))

C

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

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) + z) * y) + t
end function

Java

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

Python

def code(x, y, z, t):
	return (((x * y) + z) * y) + t

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (* (+ (* x y) z) y) t))

C

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

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) + z) * y) + t
end function

Java

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

Python

def code(x, y, z, t):
	return (((x * y) + z) * y) + t

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ t (* (+ z (* y x)) y)))

C

double code(double x, double y, double z, double t) {
	return t + ((z + (y * 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 = t + ((z + (y * x)) * y)
end function

Java

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

Python

def code(x, y, z, t):
	return t + ((z + (y * x)) * y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y + z\right) \cdot y + t \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto t + \left(z + y \cdot x\right) \cdot y \]
4. Add Preprocessing

Alternative 2: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)) (t_2 (* (fma y x z) y)))
   (if (<= t_1 -5e+105) t_2 (if (<= t_1 1e+98) (fma z y t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z + (y * x)) * y;
	double t_2 = fma(y, x, z) * y;
	double tmp;
	if (t_1 <= -5e+105) {
		tmp = t_2;
	} else if (t_1 <= 1e+98) {
		tmp = fma(z, y, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z + Float64(y * x)) * y)
	t_2 = Float64(fma(y, x, z) * y)
	tmp = 0.0
	if (t_1 <= -5e+105)
		tmp = t_2;
	elseif (t_1 <= 1e+98)
		tmp = fma(z, y, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x + z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+105], t$95$2, If[LessEqual[t$95$1, 1e+98], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -5.00000000000000046e105 or 9.99999999999999998e97 < (*.f64 (+.f64 (*.f64 x y) z) y)

   1. Initial program 99.9%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -5.00000000000000046e105 < (*.f64 (+.f64 (*.f64 x y) z) y) < 9.99999999999999998e97

   1. Initial program 99.9%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -5 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+283}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)))
   (if (<= t_1 -2e+283)
     (* (* y y) x)
     (if (<= t_1 1e+297) (fma z y t) (* (* y x) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z + (y * x)) * y;
	double tmp;
	if (t_1 <= -2e+283) {
		tmp = (y * y) * x;
	} else if (t_1 <= 1e+297) {
		tmp = fma(z, y, t);
	} else {
		tmp = (y * x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z + Float64(y * x)) * y)
	tmp = 0.0
	if (t_1 <= -2e+283)
		tmp = Float64(Float64(y * y) * x);
	elseif (t_1 <= 1e+297)
		tmp = fma(z, y, t);
	else
		tmp = Float64(Float64(y * x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+283], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e+297], N[(z * y + t), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1.99999999999999991e283

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

      4. lower-*.f64 89.5

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot x} \]

   if -1.99999999999999991e283 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1e297

   1. Initial program 99.9%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if 1e297 < (*.f64 (+.f64 (*.f64 x y) z) y)

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

      4. lower-*.f64 84.2

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -2 \cdot 10^{+283}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)) (t_2 (* (* y x) y)))
   (if (<= t_1 -2e+283) t_2 (if (<= t_1 1e+297) (fma z y t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z + (y * x)) * y;
	double t_2 = (y * x) * y;
	double tmp;
	if (t_1 <= -2e+283) {
		tmp = t_2;
	} else if (t_1 <= 1e+297) {
		tmp = fma(z, y, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z + Float64(y * x)) * y)
	t_2 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_1 <= -2e+283)
		tmp = t_2;
	elseif (t_1 <= 1e+297)
		tmp = fma(z, y, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+283], t$95$2, If[LessEqual[t$95$1, 1e+297], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1.99999999999999991e283 or 1e297 < (*.f64 (+.f64 (*.f64 x y) z) y)

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

      4. lower-*.f64 86.5

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot x \]

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot y} \]

   if -1.99999999999999991e283 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1e297

   1. Initial program 99.9%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -2 \cdot 10^{+283}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.05 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.05e+62)
   (fma z y t)
   (if (<= z 3.6e+46) (fma (* y x) y t) (fma z y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.05e+62) {
		tmp = fma(z, y, t);
	} else if (z <= 3.6e+46) {
		tmp = fma((y * x), y, t);
	} else {
		tmp = fma(z, y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.05e+62)
		tmp = fma(z, y, t);
	elseif (z <= 3.6e+46)
		tmp = fma(Float64(y * x), y, t);
	else
		tmp = fma(z, y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.05e+62], N[(z * y + t), $MachinePrecision], If[LessEqual[z, 3.6e+46], N[(N[(y * x), $MachinePrecision] * y + t), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.04999999999999999e62 or 3.5999999999999999e46 < z

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -4.04999999999999999e62 < z < 3.5999999999999999e46

   1. Initial program 99.9%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

Alternative 6: 66.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, y, t\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return fma(z, y, t)
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y + z\right) \cdot y + t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 67.1

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

5. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, t\right)} \]
6. Add Preprocessing

Alternative 7: 30.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return z * 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 = z * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y + z\right) \cdot y + t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. distribute-rgt-in N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. *-inverses N/A

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

   9. *-rgt-identity N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 59.8

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

5. Applied rewrites 59.8%

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

7. Applied rewrites 59.8%

   \[\leadsto \left(x \cdot y + z\right) \cdot y \]
8. Step-by-step derivation

9. Applied rewrites 57.1%

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

10. Taylor expanded in x around 0

    \[\leadsto y \cdot \color{blue}{z} \]
11. Step-by-step derivation

12. Applied rewrites 27.3%

    \[\leadsto z \cdot \color{blue}{y} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))