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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 7

Speedup: 1.0×

• 29.7% 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+89}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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+89) t_2 (if (<= t_1 1e+150) (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+89) {
		tmp = t_2;
	} else if (t_1 <= 1e+150) {
		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+89)
		tmp = t_2;
	elseif (t_1 <= 1e+150)
		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+89], t$95$2, If[LessEqual[t$95$1, 1e+150], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -4.99999999999999983e89 or 9.99999999999999981e149 < (*.f64 (+.f64 (*.f64 x y) z) y)

   1. Initial program 99.8%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   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 93.7

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

   5. Applied rewrites 93.7%

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

   if -4.99999999999999983e89 < (*.f64 (+.f64 (*.f64 x y) z) y) < 9.99999999999999981e149

   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 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -5 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)))
   (if (<= t_1 -1e+189)
     (* (* y x) y)
     (if (<= t_1 2e+272) (fma z y t) (* (* y y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z + (y * x)) * y;
	double tmp;
	if (t_1 <= -1e+189) {
		tmp = (y * x) * y;
	} else if (t_1 <= 2e+272) {
		tmp = fma(z, y, t);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z + Float64(y * x)) * y)
	tmp = 0.0
	if (t_1 <= -1e+189)
		tmp = Float64(Float64(y * x) * y);
	elseif (t_1 <= 2e+272)
		tmp = fma(z, y, t);
	else
		tmp = Float64(Float64(y * y) * x);
	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, -1e+189], N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], N[(z * y + t), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1e189

   1. Initial program 99.8%

      \[\left(x \cdot y + z\right) \cdot y + t \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.8%

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

   if -1e189 < (*.f64 (+.f64 (*.f64 x y) z) y) < 2.0000000000000001e272

   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 83.7

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

   5. Applied rewrites 83.7%

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

   if 2.0000000000000001e272 < (*.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 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + y \cdot x\right) \cdot y\\ t_2 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+189}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z (* y x)) y)) (t_2 (* (* y x) y)))
   (if (<= t_1 -1e+189) t_2 (if (<= t_1 2e+272) (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 <= -1e+189) {
		tmp = t_2;
	} else if (t_1 <= 2e+272) {
		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 <= -1e+189)
		tmp = t_2;
	elseif (t_1 <= 2e+272)
		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, -1e+189], t$95$2, If[LessEqual[t$95$1, 2e+272], N[(z * y + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 (*.f64 x y) z) y) < -1e189 or 2.0000000000000001e272 < (*.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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.5%

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

   if -1e189 < (*.f64 (+.f64 (*.f64 x y) z) y) < 2.0000000000000001e272

   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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + y \cdot x\right) \cdot y \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(z + y \cdot x\right) \cdot y \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e-17)
   (fma z y t)
   (if (<= z 1.15e+74) (fma (* y x) y t) (fma z y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e-17) {
		tmp = fma(z, y, t);
	} else if (z <= 1.15e+74) {
		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 <= -3.5e-17)
		tmp = fma(z, y, t);
	elseif (z <= 1.15e+74)
		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, -3.5e-17], N[(z * y + t), $MachinePrecision], If[LessEqual[z, 1.15e+74], N[(N[(y * x), $MachinePrecision] * y + t), $MachinePrecision], N[(z * y + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000002e-17 or 1.1499999999999999e74 < 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 -3.5000000000000002e-17 < z < 1.1499999999999999e74

   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 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 6: 66.5% 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 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 67.8

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

5. Applied rewrites 67.8%

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

Alternative 7: 29.7% 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 62.0

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

5. Applied rewrites 62.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 30.3%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))