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

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 7

Speedup: 1.0×

• 28.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+139}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -1e+139) t_2 (if (<= t_1 2e+135) (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 <= -1e+139) {
		tmp = t_2;
	} else if (t_1 <= 2e+135) {
		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 <= -1e+139)
		tmp = t_2;
	elseif (t_1 <= 2e+135)
		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, -1e+139], t$95$2, If[LessEqual[t$95$1, 2e+135], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1.00000000000000003e139 or 1.99999999999999992e135 < (*.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 t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1.00000000000000003e139 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1.99999999999999992e135

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)))
   (if (<= t_1 -2e+255)
     (* (* y y) x)
     (if (<= t_1 5e+188) (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+255) {
		tmp = (y * y) * x;
	} else if (t_1 <= 5e+188) {
		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+255)
		tmp = Float64(Float64(y * y) * x);
	elseif (t_1 <= 5e+188)
		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+255], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e+188], 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) < -1.99999999999999998e255

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot {y}^{2}} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -1.99999999999999998e255 < (*.f64 (+.f64 (*.f64 x y) z) y) < 5.0000000000000001e188

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 86.3

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

   5. Applied rewrites 86.3%

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

   if 5.0000000000000001e188 < (*.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

      \[\leadsto \color{blue}{x \cdot {y}^{2}} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 85.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)) (t_2 (* (* y x) y)))
   (if (<= t_1 -2e+255) t_2 (if (<= t_1 5e+188) (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+255) {
		tmp = t_2;
	} else if (t_1 <= 5e+188) {
		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+255)
		tmp = t_2;
	elseif (t_1 <= 5e+188)
		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+255], t$95$2, If[LessEqual[t$95$1, 5e+188], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1.99999999999999998e255 or 5.0000000000000001e188 < (*.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

      \[\leadsto \color{blue}{x \cdot {y}^{2}} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 84.4%

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

   if -1.99999999999999998e255 < (*.f64 (+.f64 (*.f64 x y) z) y) < 5.0000000000000001e188

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 86.3

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

   5. Applied rewrites 86.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+96}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+135}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)))
   (if (<= t_1 -2e+96) (* z y) (if (<= t_1 2e+135) (* 1.0 t) (* z 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+96) {
		tmp = z * y;
	} else if (t_1 <= 2e+135) {
		tmp = 1.0 * t;
	} else {
		tmp = z * 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) :: t_1
    real(8) :: tmp
    t_1 = (z + (y * x)) * y
    if (t_1 <= (-2d+96)) then
        tmp = z * y
    else if (t_1 <= 2d+135) then
        tmp = 1.0d0 * t
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z + (y * x)) * y;
	double tmp;
	if (t_1 <= -2e+96) {
		tmp = z * y;
	} else if (t_1 <= 2e+135) {
		tmp = 1.0 * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z + (y * x)) * y
	tmp = 0
	if t_1 <= -2e+96:
		tmp = z * y
	elif t_1 <= 2e+135:
		tmp = 1.0 * t
	else:
		tmp = z * 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+96)
		tmp = Float64(z * y);
	elseif (t_1 <= 2e+135)
		tmp = Float64(1.0 * t);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z + (y * x)) * y;
	tmp = 0.0;
	if (t_1 <= -2e+96)
		tmp = z * y;
	elseif (t_1 <= 2e+135)
		tmp = 1.0 * t;
	else
		tmp = z * y;
	end
	tmp_2 = 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+96], N[(z * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+135], N[(1.0 * t), $MachinePrecision], N[(z * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -2.0000000000000001e96 or 1.99999999999999992e135 < (*.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.4

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

   5. Applied rewrites 32.4%

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

   if -2.0000000000000001e96 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1.99999999999999992e135

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot y + z\right) \cdot y\right)}^{3} + {t}^{3}}{\left(\left(x \cdot y + z\right) \cdot y\right) \cdot \left(\left(x \cdot y + z\right) \cdot y\right) + \left(t \cdot t - \left(\left(x \cdot y + z\right) \cdot y\right) \cdot t\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot y + z\right) \cdot y\right) \cdot \left(\left(x \cdot y + z\right) \cdot y\right) + \left(t \cdot t - \left(\left(x \cdot y + z\right) \cdot y\right) \cdot t\right)}{{\left(\left(x \cdot y + z\right) \cdot y\right)}^{3} + {t}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot y + z\right) \cdot y\right) \cdot \left(\left(x \cdot y + z\right) \cdot y\right) + \left(t \cdot t - \left(\left(x \cdot y + z\right) \cdot y\right) \cdot t\right)}{{\left(\left(x \cdot y + z\right) \cdot y\right)}^{3} + {t}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot y + z\right) \cdot y\right)}^{3} + {t}^{3}}{\left(\left(x \cdot y + z\right) \cdot y\right) \cdot \left(\left(x \cdot y + z\right) \cdot y\right) + \left(t \cdot t - \left(\left(x \cdot y + z\right) \cdot y\right) \cdot t\right)}}}} \]

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.4

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto 1 \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 78.5%

       \[\leadsto 1 \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -2 \cdot 10^{+96}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% 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 y 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 60.8

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

5. Applied rewrites 60.8%

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

Alternative 7: 29.5% 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 z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Applied rewrites 25.3%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))