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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 28.9% 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

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y + z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+93} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+195}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ (* x y) z) y)))
   (if (or (<= t_1 -2e+93) (not (<= t_1 5e+195)))
     (* (fma y x z) y)
     (fma z y t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) + z) * y;
	double tmp;
	if ((t_1 <= -2e+93) || !(t_1 <= 5e+195)) {
		tmp = fma(y, x, z) * y;
	} else {
		tmp = fma(z, y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) + z) * y)
	tmp = 0.0
	if ((t_1 <= -2e+93) || !(t_1 <= 5e+195))
		tmp = Float64(fma(y, x, z) * y);
	else
		tmp = fma(z, y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+93], N[Not[LessEqual[t$95$1, 5e+195]], $MachinePrecision]], N[(N[(y * x + z), $MachinePrecision] * y), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -2.00000000000000009e93 or 4.9999999999999998e195 < (*.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. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

      12. unpow2 N/A

          \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

      13. associate-*l* N/A

          \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      15. +-lft-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. cancel-sign-sub N/A

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

      18. mul-1-neg N/A

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

   5. Applied rewrites 96.3%

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

   if -2.00000000000000009e93 < (*.f64 (+.f64 (*.f64 x y) z) y) < 4.9999999999999998e195

   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 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z\right) \cdot y \leq -2 \cdot 10^{+93} \lor \neg \left(\left(x \cdot y + z\right) \cdot y \leq 5 \cdot 10^{+195}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y + z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+292} \lor \neg \left(t\_1 \leq 10^{+218}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ (* x y) z) y)))
   (if (or (<= t_1 -2e+292) (not (<= t_1 1e+218))) (* (* x y) y) (fma z y t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) + z) * y;
	double tmp;
	if ((t_1 <= -2e+292) || !(t_1 <= 1e+218)) {
		tmp = (x * y) * y;
	} else {
		tmp = fma(z, y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) + z) * y)
	tmp = 0.0
	if ((t_1 <= -2e+292) || !(t_1 <= 1e+218))
		tmp = Float64(Float64(x * y) * y);
	else
		tmp = fma(z, y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+292], N[Not[LessEqual[t$95$1, 1e+218]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -2e292 or 1.00000000000000008e218 < (*.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 y around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

      12. unpow2 N/A

          \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

      13. associate-*l* N/A

          \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      15. +-lft-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. cancel-sign-sub N/A

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

      18. mul-1-neg N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.1%

      \[\leadsto \left(x \cdot y\right) \cdot y \]

   if -2e292 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1.00000000000000008e218

   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 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z\right) \cdot y \leq -2 \cdot 10^{+292} \lor \neg \left(\left(x \cdot y + z\right) \cdot y \leq 10^{+218}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y + z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ (* x y) z) y)))
   (if (<= t_1 -2e+292)
     (* (* y y) x)
     (if (<= t_1 1e+218) (fma z y t) (* (* x y) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) + z) * y;
	double tmp;
	if (t_1 <= -2e+292) {
		tmp = (y * y) * x;
	} else if (t_1 <= 1e+218) {
		tmp = fma(z, y, t);
	} else {
		tmp = (x * y) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) + z) * y)
	tmp = 0.0
	if (t_1 <= -2e+292)
		tmp = Float64(Float64(y * y) * x);
	elseif (t_1 <= 1e+218)
		tmp = fma(z, y, t);
	else
		tmp = Float64(Float64(x * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+292], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e+218], N[(z * y + t), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -2e292

   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 78.2

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

   5. Applied rewrites 78.2%

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

   if -2e292 < (*.f64 (+.f64 (*.f64 x y) z) y) < 1.00000000000000008e218

   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 82.4

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

   5. Applied rewrites 82.4%

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

   if 1.00000000000000008e218 < (*.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. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

      12. unpow2 N/A

          \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

      13. associate-*l* N/A

          \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      15. +-lft-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. cancel-sign-sub N/A

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

      18. mul-1-neg N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.3%

      \[\leadsto \left(x \cdot y\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(x \cdot y\right) \cdot y\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.95e+147)
   (fma z y t)
   (if (<= z -1.3e+14)
     (fma z y (* (* x y) y))
     (if (<= z 3.9e+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 <= -1.95e+147) {
		tmp = fma(z, y, t);
	} else if (z <= -1.3e+14) {
		tmp = fma(z, y, ((x * y) * y));
	} else if (z <= 3.9e+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 <= -1.95e+147)
		tmp = fma(z, y, t);
	elseif (z <= -1.3e+14)
		tmp = fma(z, y, Float64(Float64(x * y) * y));
	elseif (z <= 3.9e+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, -1.95e+147], N[(z * y + t), $MachinePrecision], If[LessEqual[z, -1.3e+14], N[(z * y + N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.9e+46], N[(N[(y * x), $MachinePrecision] * y + t), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95000000000000008e147 or 3.89999999999999995e46 < 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 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.95000000000000008e147 < z < -1.3e14

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   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. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

      12. unpow2 N/A

          \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

      13. associate-*l* N/A

          \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      15. +-lft-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. cancel-sign-sub N/A

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

      18. mul-1-neg N/A

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   if -1.3e14 < z < 3.89999999999999995e46

   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 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 6: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.95e+147)
   (fma z y t)
   (if (<= z -1.3e+14)
     (* (fma y x z) y)
     (if (<= z 3.9e+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 <= -1.95e+147) {
		tmp = fma(z, y, t);
	} else if (z <= -1.3e+14) {
		tmp = fma(y, x, z) * y;
	} else if (z <= 3.9e+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 <= -1.95e+147)
		tmp = fma(z, y, t);
	elseif (z <= -1.3e+14)
		tmp = Float64(fma(y, x, z) * y);
	elseif (z <= 3.9e+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, -1.95e+147], N[(z * y + t), $MachinePrecision], If[LessEqual[z, -1.3e+14], N[(N[(y * x + z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 3.9e+46], N[(N[(y * x), $MachinePrecision] * y + t), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95000000000000008e147 or 3.89999999999999995e46 < 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 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.95000000000000008e147 < z < -1.3e14

   1. Initial program 100.0%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   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. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      7. mul0-rgt N/A

         \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

      12. unpow2 N/A

          \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

      13. associate-*l* N/A

          \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

      15. +-lft-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. cancel-sign-sub N/A

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

      18. mul-1-neg N/A

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

   5. Applied rewrites 94.0%

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

   if -1.3e14 < z < 3.89999999999999995e46

   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 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 7: 66.5% 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 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 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 8: 29.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + z\right) \cdot y + t \]
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. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-lft-out N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - {y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)} \]

   6. unsub-neg N/A

      \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

   7. mul0-rgt N/A

      \[\leadsto \color{blue}{y \cdot 0} + \left(\mathsf{neg}\left({y}^{2} \cdot \left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right) \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto y \cdot 0 + \color{blue}{{y}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot x + -1 \cdot \frac{z}{y}\right)\right)\right)} \]

   9. distribute-lft-out N/A

      \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x + \frac{z}{y}\right)}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto y \cdot 0 + {y}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x + \frac{z}{y}\right)\right)\right)}\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto y \cdot 0 + {y}^{2} \cdot \color{blue}{\left(x + \frac{z}{y}\right)} \]

   12. unpow2 N/A

       \[\leadsto y \cdot 0 + \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{z}{y}\right) \]

   13. associate-*l* N/A

       \[\leadsto y \cdot 0 + \color{blue}{y \cdot \left(y \cdot \left(x + \frac{z}{y}\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{y \cdot \left(0 + y \cdot \left(x + \frac{z}{y}\right)\right)} \]

   15. +-lft-identity N/A

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

   16. distribute-rgt-in N/A

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

   17. cancel-sign-sub N/A

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

   18. mul-1-neg N/A

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.5%

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

Reproduce

herbie shell --seed 2024324 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))