Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.4% → 99.7%

Time: 3.2s

Alternatives: 6

Speedup: 1.0×

• 22.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-11} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -2e-11) (not (<= (* x y) 5e-63)))
   (fma y x (* t z))
   (* (- z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -2e-11) || !((x * y) <= 5e-63)) {
		tmp = fma(y, x, (t * z));
	} else {
		tmp = -z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -2e-11) || !(Float64(x * y) <= 5e-63))
		tmp = fma(y, x, Float64(t * z));
	else
		tmp = Float64(Float64(-z) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-11], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-63]], $MachinePrecision]], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], N[((-z) * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999988e-11 or 5.0000000000000002e-63 < (*.f64 x y)

   1. Initial program 97.1%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(0 - t\right)} \cdot z\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{{0}^{3} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)}} \cdot z\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\color{blue}{0} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\color{blue}{\mathsf{neg}\left({t}^{3}\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(t \cdot t\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\left(\color{blue}{\left(-t\right)} \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\left(\left(-t\right) \cdot \color{blue}{\left(-t\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      11. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(-t\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-t\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      12. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(-t\right)}^{3}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(t\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      14. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({t}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{\left(0 - {t}^{3}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {t}^{3}\right)\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(\frac{{0}^{3} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)}\right)\right)} \cdot z\right) \]

      18. flip3-- N/A

          \[\leadsto \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{\left(0 - t\right)}\right)\right) \cdot z\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg 70.9

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

   6. Applied rewrites 70.9%

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

   if -1.99999999999999988e-11 < (*.f64 x y) < 5.0000000000000002e-63

   1. Initial program 100.0%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot t \]

      5. lower-neg.f64 86.6

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\left(-z\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-11} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-63}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -2e-11)
   (fma y x (* t z))
   (if (<= (* x y) 5e-63) (* (- z) t) (fma z t (* y x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -2e-11) {
		tmp = fma(y, x, (t * z));
	} else if ((x * y) <= 5e-63) {
		tmp = -z * t;
	} else {
		tmp = fma(z, t, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -2e-11)
		tmp = fma(y, x, Float64(t * z));
	elseif (Float64(x * y) <= 5e-63)
		tmp = Float64(Float64(-z) * t);
	else
		tmp = fma(z, t, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-11], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-63], N[((-z) * t), $MachinePrecision], N[(z * t + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999988e-11

   1. Initial program 96.4%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(0 - t\right)} \cdot z\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{{0}^{3} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)}} \cdot z\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\color{blue}{0} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\color{blue}{\mathsf{neg}\left({t}^{3}\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(t \cdot t\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\left(\color{blue}{\left(-t\right)} \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\left(\left(-t\right) \cdot \color{blue}{\left(-t\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      11. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(-t\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-t\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      12. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{{\left(-t\right)}^{3}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(t\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      14. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({t}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\color{blue}{\left(0 - {t}^{3}\right)}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, x, \frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {t}^{3}\right)\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \cdot z\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(\frac{{0}^{3} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)}\right)\right)} \cdot z\right) \]

      18. flip3-- N/A

          \[\leadsto \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{\left(0 - t\right)}\right)\right) \cdot z\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg 72.5

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

   6. Applied rewrites 72.5%

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

   if -1.99999999999999988e-11 < (*.f64 x y) < 5.0000000000000002e-63

   1. Initial program 100.0%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot t \]

      5. lower-neg.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 5.0000000000000002e-63 < (*.f64 x y)

   1. Initial program 97.6%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 98.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - z}, t, y \cdot x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}}, t, y \cdot x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\color{blue}{\left(-z\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\left(-z\right) \cdot \color{blue}{\left(-z\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      11. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(-z\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-z\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      12. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(-z\right)}^{3}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      14. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({z}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(0 - {z}^{3}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, t, y \cdot x\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)}, t, y \cdot x\right) \]

      18. flip3-- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(0 - z\right)}\right), t, y \cdot x\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg 71.0

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

   6. Applied rewrites 71.0%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 52.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot t \]

   5. lower-neg.f64 55.5

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

5. Applied rewrites 55.5%

   \[\leadsto \color{blue}{\left(-z\right) \cdot t} \]
6. Add Preprocessing

Alternative 6: 3.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot t \]

   5. lower-neg.f64 55.5

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

5. Applied rewrites 55.5%

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

7. Applied rewrites 4.4%

   \[\leadsto z \cdot \color{blue}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024317 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))