Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Percentage Accurate: 99.5% → 99.7%

Time: 11.2s

Alternatives: 15

Speedup: 1.7×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (- y x) 4.0 (fma (* -6.0 z) (- y x) x)))

C

double code(double x, double y, double z) {
	return fma((y - x), 4.0, fma((-6.0 * z), (y - x), x));
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), 4.0, fma(Float64(-6.0 * z), Float64(y - x), x))
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * 4.0 + N[(N[(-6.0 * z), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift--.f64 N/A

      \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

   5. sub-neg N/A

      \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

   7. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   9. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

   18. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]
5. Add Preprocessing

Alternative 2: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 0.66666666666:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.6666666666666669:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 0.66666666666)
     t_1
     (if (<= t_0 0.6666666666666669)
       (fma (- y x) 4.0 x)
       (if (<= t_0 1e+252) (* (fma 6.0 z -3.0) x) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= 0.66666666666) {
		tmp = t_1;
	} else if (t_0 <= 0.6666666666666669) {
		tmp = fma((y - x), 4.0, x);
	} else if (t_0 <= 1e+252) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= 0.66666666666)
		tmp = t_1;
	elseif (t_0 <= 0.6666666666666669)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (t_0 <= 1e+252)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.66666666666], t$95$1, If[LessEqual[t$95$0, 0.6666666666666669], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[t$95$0, 1e+252], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666666659999962 or 1.0000000000000001e252 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. sub-neg N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \cdot y \]

      5. mul-1-neg N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} + \color{blue}{-1 \cdot z}\right)\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(-1 \cdot z\right) + 6 \cdot \frac{2}{3}\right)} \cdot y \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}\right) \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(-6 \cdot z + \color{blue}{4}\right) \cdot y \]

      11. lower-fma.f64 64.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if 0.666666666659999962 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666666666666852

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 0.666666666666666852 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1.0000000000000001e252

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(\left(\frac{2}{3} - z\right) \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot \left(\frac{2}{3} - z\right)\right)} + 1 \cdot x \]

      5. metadata-eval N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{\left(-1 \cdot -1\right)} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{-1 \cdot \left(-1 \cdot x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right)\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot x\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + -1 \cdot x\right)\right)} \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(6 \cdot \left(\frac{2}{3} - z\right) + -1\right)}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \left(6 \cdot \left(\frac{2}{3} - z\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)}\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right) \cdot x} \]

      18. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right) \cdot x} \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z, -3\right) \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 0.66666666666:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+252}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 0.66666666666)
     t_1
     (if (<= t_0 40000000.0)
       (fma (- y x) 4.0 x)
       (if (<= t_0 1e+252) (* (* x z) 6.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= 0.66666666666) {
		tmp = t_1;
	} else if (t_0 <= 40000000.0) {
		tmp = fma((y - x), 4.0, x);
	} else if (t_0 <= 1e+252) {
		tmp = (x * z) * 6.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= 0.66666666666)
		tmp = t_1;
	elseif (t_0 <= 40000000.0)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (t_0 <= 1e+252)
		tmp = Float64(Float64(x * z) * 6.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.66666666666], t$95$1, If[LessEqual[t$95$0, 40000000.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[t$95$0, 1e+252], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666666659999962 or 1.0000000000000001e252 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. sub-neg N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \cdot y \]

      5. mul-1-neg N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} + \color{blue}{-1 \cdot z}\right)\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(-1 \cdot z\right) + 6 \cdot \frac{2}{3}\right)} \cdot y \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}\right) \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(-6 \cdot z + \color{blue}{4}\right) \cdot y \]

      11. lower-fma.f64 64.2

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if 0.666666666659999962 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 4e7

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 98.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 4e7 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1.0000000000000001e252

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 99.1

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 66.0%

       \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(y \cdot z\right) \cdot -6\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+252}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* y z) -6.0)))
   (if (<= t_0 -10000.0)
     t_1
     (if (<= t_0 40000000.0)
       (fma (- y x) 4.0 x)
       (if (<= t_0 1e+252) (* (* x z) 6.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = (y * z) * -6.0;
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = t_1;
	} else if (t_0 <= 40000000.0) {
		tmp = fma((y - x), 4.0, x);
	} else if (t_0 <= 1e+252) {
		tmp = (x * z) * 6.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(y * z) * -6.0)
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = t_1;
	elseif (t_0 <= 40000000.0)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (t_0 <= 1e+252)
		tmp = Float64(Float64(x * z) * 6.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], t$95$1, If[LessEqual[t$95$0, 40000000.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[t$95$0, 1e+252], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4 or 1.0000000000000001e252 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 98.4

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.4%

       \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{-6} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 4e7

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 96.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 4e7 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1.0000000000000001e252

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 99.1

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 66.0%

       \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x - y\right) \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -10000.0)
     (* (- y x) (* -6.0 z))
     (if (<= t_0 1.0) (fma -3.0 x (* 4.0 y)) (fma (* (- x y) z) 6.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = (y - x) * (-6.0 * z);
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = fma(((x - y) * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(Float64(y - x) * Float64(-6.0 * z));
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = fma(Float64(Float64(x - y) * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(N[(y - x), $MachinePrecision] * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 98.1

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.1%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(-6 \cdot z\right)} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x + 4 \cdot \left(y + \color{blue}{-1 \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(-1 \cdot x + y\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{\left(4 \cdot \left(-1 \cdot x\right) + 4 \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + 4 \cdot y\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot x\right)\right)} + 4 \cdot y\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot x} + 4 \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \left(\color{blue}{-4} \cdot x + 4 \cdot y\right) \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      13. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot 6} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right), 6, x\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

      10. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

      12. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0.6666666666666666} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)}, 6, x\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot \left(y - x\right)\right) \cdot z}, 6, x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot \left(y - x\right)\right) \cdot z}, 6, x\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right)} \cdot z, 6, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right)\right) \cdot z, 6, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)}\right)\right) \cdot z, 6, x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)} \cdot z, 6, x\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right)} \cdot z, 6, x\right) \]

      9. remove-double-neg N/A

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

      10. lower--.f64 98.6

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

   7. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x - y\right) \cdot z}, 6, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -10000 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (or (<= t_0 -10000.0) (not (<= t_0 1.0)))
     (* (* (- y x) z) -6.0)
     (fma -3.0 x (* 4.0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if ((t_0 <= -10000.0) || !(t_0 <= 1.0)) {
		tmp = ((y - x) * z) * -6.0;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if ((t_0 <= -10000.0) || !(t_0 <= 1.0))
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10000.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 98.3

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x + 4 \cdot \left(y + \color{blue}{-1 \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(-1 \cdot x + y\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{\left(4 \cdot \left(-1 \cdot x\right) + 4 \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + 4 \cdot y\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot x\right)\right)} + 4 \cdot y\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot x} + 4 \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \left(\color{blue}{-4} \cdot x + 4 \cdot y\right) \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      13. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -10000 \lor \neg \left(\frac{2}{3} - z \leq 1\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -10000.0)
     (* (- y x) (* -6.0 z))
     (if (<= t_0 1.0) (fma -3.0 x (* 4.0 y)) (* (* (- y x) z) -6.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = (y - x) * (-6.0 * z);
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = ((y - x) * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(Float64(y - x) * Float64(-6.0 * z));
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(N[(y - x), $MachinePrecision] * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 98.1

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.1%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(-6 \cdot z\right)} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x + 4 \cdot \left(y + \color{blue}{-1 \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(-1 \cdot x + y\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{\left(4 \cdot \left(-1 \cdot x\right) + 4 \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + 4 \cdot y\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot x\right)\right)} + 4 \cdot y\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot x} + 4 \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \left(\color{blue}{-4} \cdot x + 4 \cdot y\right) \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      13. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 98.6

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -10000.0)
     (* (* -6.0 (- y x)) z)
     (if (<= t_0 1.0) (fma -3.0 x (* 4.0 y)) (* (* (- y x) z) -6.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = (-6.0 * (y - x)) * z;
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = ((y - x) * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 98.1

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x + 4 \cdot \left(y + \color{blue}{-1 \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(-1 \cdot x + y\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{\left(4 \cdot \left(-1 \cdot x\right) + 4 \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + 4 \cdot y\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot x\right)\right)} + 4 \cdot y\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot x} + 4 \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \left(\color{blue}{-4} \cdot x + 4 \cdot y\right) \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      13. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 98.6

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -10000 \lor \neg \left(t\_0 \leq 40000000\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (or (<= t_0 -10000.0) (not (<= t_0 40000000.0)))
     (* (* x z) 6.0)
     (fma (- y x) 4.0 x))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if ((t_0 <= -10000.0) || !(t_0 <= 40000000.0)) {
		tmp = (x * z) * 6.0;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if ((t_0 <= -10000.0) || !(t_0 <= 40000000.0))
		tmp = Float64(Float64(x * z) * 6.0);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10000.0], N[Not[LessEqual[t$95$0, 40000000.0]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e4 or 4e7 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right)} \cdot z \]

      5. lower--.f64 98.6

         \[\leadsto \left(-6 \cdot \color{blue}{\left(y - x\right)}\right) \cdot z \]

   7. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right)\right) \cdot z} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 50.9%

       \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]

   if -1e4 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 4e7

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 96.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 96.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -10000 \lor \neg \left(\frac{2}{3} - z \leq 40000000\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= z -1.65e+252)
     t_0
     (if (<= z -1.05e-19)
       (* (fma 6.0 z -3.0) x)
       (if (<= z 7.8e-19) (fma -3.0 x (* 4.0 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (z <= -1.65e+252) {
		tmp = t_0;
	} else if (z <= -1.05e-19) {
		tmp = fma(6.0, z, -3.0) * x;
	} else if (z <= 7.8e-19) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (z <= -1.65e+252)
		tmp = t_0;
	elseif (z <= -1.05e-19)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	elseif (z <= 7.8e-19)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.65e+252], t$95$0, If[LessEqual[z, -1.05e-19], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 7.8e-19], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.65e252 or 7.7999999999999999e-19 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. sub-neg N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \cdot y \]

      5. mul-1-neg N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} + \color{blue}{-1 \cdot z}\right)\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(-1 \cdot z\right) + 6 \cdot \frac{2}{3}\right)} \cdot y \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}\right) \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(-6 \cdot z + \color{blue}{4}\right) \cdot y \]

      11. lower-fma.f64 64.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if -1.65e252 < z < -1.0499999999999999e-19

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(\left(\frac{2}{3} - z\right) \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot \left(\frac{2}{3} - z\right)\right)} + 1 \cdot x \]

      5. metadata-eval N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{\left(-1 \cdot -1\right)} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{-1 \cdot \left(-1 \cdot x\right)} \]

      7. mul-1-neg N/A

         \[\leadsto -6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(x \cdot \left(\frac{2}{3} - z\right)\right)\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot x\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + -1 \cdot x\right)\right)} \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(6 \cdot \left(\frac{2}{3} - z\right) + -1\right)}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \left(6 \cdot \left(\frac{2}{3} - z\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)}\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right) \cdot x} \]

      18. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \left(\frac{2}{3} - z\right) - 1\right)\right)\right) \cdot x} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z, -3\right) \cdot x} \]

   if -1.0499999999999999e-19 < z < 7.7999999999999999e-19

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

      5. sub-neg N/A

         \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + x\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6\right) \cdot \left(y - x\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot 6, y - x, x\right)}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(-6 \cdot z, y - x, x\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x + 4 \cdot \left(y + \color{blue}{-1 \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(-1 \cdot x + y\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{\left(4 \cdot \left(-1 \cdot x\right) + 4 \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + 4 \cdot y\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot x\right)\right)} + 4 \cdot y\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot x} + 4 \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \left(\color{blue}{-4} \cdot x + 4 \cdot y\right) \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      13. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 99.9%

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

Alternative 11: 38.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+56} \lor \neg \left(y \leq 2.95 \cdot 10^{+54}\right):\\ \;\;\;\;4 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.7e+56) (not (<= y 2.95e+54))) (* 4.0 y) (* -3.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e+56) || !(y <= 2.95e+54)) {
		tmp = 4.0 * y;
	} else {
		tmp = -3.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.7d+56)) .or. (.not. (y <= 2.95d+54))) then
        tmp = 4.0d0 * y
    else
        tmp = (-3.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e+56) || !(y <= 2.95e+54)) {
		tmp = 4.0 * y;
	} else {
		tmp = -3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.7e+56) or not (y <= 2.95e+54):
		tmp = 4.0 * y
	else:
		tmp = -3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.7e+56) || !(y <= 2.95e+54))
		tmp = Float64(4.0 * y);
	else
		tmp = Float64(-3.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.7e+56) || ~((y <= 2.95e+54)))
		tmp = 4.0 * y;
	else
		tmp = -3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.7e+56], N[Not[LessEqual[y, 2.95e+54]], $MachinePrecision]], N[(4.0 * y), $MachinePrecision], N[(-3.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7000000000000001e56 or 2.9499999999999999e54 < y

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 55.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.5%

      \[\leadsto 4 \cdot \color{blue}{y} \]

   if -2.7000000000000001e56 < y < 2.9499999999999999e54

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 48.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto -3 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.3%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+56} \lor \neg \left(y \leq 2.95 \cdot 10^{+54}\right):\\ \;\;\;\;4 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, \left(0.6666666666666666 - z\right) \cdot 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (- y x) (* (- 0.6666666666666666 z) 6.0) x))

C

double code(double x, double y, double z) {
	return fma((y - x), ((0.6666666666666666 - z) * 6.0), x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), Float64(Float64(0.6666666666666666 - z) * 6.0), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * N[(N[(0.6666666666666666 - z), $MachinePrecision] * 6.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)} + x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot 6} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right), 6, x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   12. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0.6666666666666666} - z\right) \cdot \left(y - x\right), 6, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right)} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)\right) \cdot 6 + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)\right)} \cdot 6 + x \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

   5. lift--.f64 N/A

      \[\leadsto \left(\frac{2}{3} - z\right) \cdot \left(\color{blue}{\left(y - x\right)} \cdot 6\right) + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   7. lift--.f64 N/A

      \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

   8. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x \]

   9. lift--.f64 N/A

      \[\leadsto \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} - z\right)} + x \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right)} \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\left(\frac{2}{3} - z\right) \cdot 6}, x\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\left(\frac{2}{3} - z\right) \cdot 6}, x\right) \]

   14. metadata-eval 99.7

       \[\leadsto \mathsf{fma}\left(y - x, \left(\color{blue}{0.6666666666666666} - z\right) \cdot 6, x\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \left(0.6666666666666666 - z\right) \cdot 6, x\right)} \]
7. Add Preprocessing

Alternative 13: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (* (- 0.6666666666666666 z) (- y x)) 6.0 x))

C

double code(double x, double y, double z) {
	return fma(((0.6666666666666666 - z) * (y - x)), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(0.6666666666666666 - z) * Float64(y - x)), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)} + x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot 6} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right), 6, x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   12. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0.6666666666666666} - z\right) \cdot \left(y - x\right), 6, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right)} \]
5. Add Preprocessing

Alternative 14: 50.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, 4, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) 4.0 x))

C

double code(double x, double y, double z) {
	return fma((y - x), 4.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), 4.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 51.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]
6. Add Preprocessing

Alternative 15: 25.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ -3 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -3.0 x))

C

double code(double x, double y, double z) {
	return -3.0 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-3.0d0) * x
end function

Java

public static double code(double x, double y, double z) {
	return -3.0 * x;
}

Python

def code(x, y, z):
	return -3.0 * x

Julia

function code(x, y, z)
	return Float64(-3.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -3.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(-3.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 51.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto -3 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 28.0%

   \[\leadsto -3 \cdot \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))