Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Percentage Accurate: 70.5% → 76.8%

Time: 15.8s

Alternatives: 7

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot -0.3333333333333333}{b}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* 2.0 (sqrt x)) (cos y) (/ (* a -0.3333333333333333) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((2.0 * sqrt(x)), cos(y), ((a * -0.3333333333333333) / b));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(2.0 * sqrt(x)), cos(y), Float64(Float64(a * -0.3333333333333333) / b))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 81.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Simplified 81.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   3. metadata-eval N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}}\right) \]

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

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b}\right) \]

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

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b}\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b}\right) \]

   15. lower-*.f64 81.4

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot -0.3333333333333333}}{b}\right) \]

8. Simplified 81.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot -0.3333333333333333}{b}\right)} \]
9. Add Preprocessing

Alternative 2: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-93}:\\ \;\;\;\;t\_1 \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* b 3.0))))
   (if (<= t_2 -2e+18)
     (fma -0.3333333333333333 (/ a b) t_1)
     (if (<= t_2 1e-93) (* t_1 (cos y)) (- t_1 t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double t_2 = a / (b * 3.0);
	double tmp;
	if (t_2 <= -2e+18) {
		tmp = fma(-0.3333333333333333, (a / b), t_1);
	} else if (t_2 <= 1e-93) {
		tmp = t_1 * cos(y);
	} else {
		tmp = t_1 - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * sqrt(x))
	t_2 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (t_2 <= -2e+18)
		tmp = fma(-0.3333333333333333, Float64(a / b), t_1);
	elseif (t_2 <= 1e-93)
		tmp = Float64(t_1 * cos(y));
	else
		tmp = Float64(t_1 - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+18], N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e-93], N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2e18

   1. Initial program 82.7%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 92.6

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Simplified 92.6%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b}\right) \]

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

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b}\right) \]

      15. lower-*.f64 92.6

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot -0.3333333333333333}}{b}\right) \]

   8. Simplified 92.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot -0.3333333333333333}{b}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b} + 2 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{a}{b}, 2 \cdot \sqrt{x}\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{a}{b}}, 2 \cdot \sqrt{x}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \frac{a}{b}, \color{blue}{2 \cdot \sqrt{x}}\right) \]

       4. lower-sqrt.f64 90.1

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, 2 \cdot \color{blue}{\sqrt{x}}\right) \]

   11. Simplified 90.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, 2 \cdot \sqrt{x}\right)} \]

   if -2e18 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 9.999999999999999e-94

   1. Initial program 61.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 60.8

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Simplified 60.8%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right)} \cdot \cos y \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\sqrt{x}}\right) \cdot \cos y \]

      5. lower-cos.f64 59.8

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} \]

   8. Simplified 59.8%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \]

   if 9.999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 77.7%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 90.1

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Simplified 90.1%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

      2. lower-sqrt.f64 82.8

         \[\leadsto 2 \cdot \color{blue}{\sqrt{x}} - \frac{a}{b \cdot 3} \]

   8. Simplified 82.8%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 57.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \frac{a \cdot -0.3333333333333333}{b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (/ (* a -0.3333333333333333) b)))
   (if (<= t_1 -2e+18)
     t_2
     (if (<= t_1 5e-109) (* (sqrt x) (- 2.0 (* y y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = (a * -0.3333333333333333) / b;
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = t_2;
	} else if (t_1 <= 5e-109) {
		tmp = sqrt(x) * (2.0 - (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = (a * (-0.3333333333333333d0)) / b
    if (t_1 <= (-2d+18)) then
        tmp = t_2
    else if (t_1 <= 5d-109) then
        tmp = sqrt(x) * (2.0d0 - (y * y))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = (a * -0.3333333333333333) / b;
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = t_2;
	} else if (t_1 <= 5e-109) {
		tmp = Math.sqrt(x) * (2.0 - (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = (a * -0.3333333333333333) / b
	tmp = 0
	if t_1 <= -2e+18:
		tmp = t_2
	elif t_1 <= 5e-109:
		tmp = math.sqrt(x) * (2.0 - (y * y))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(Float64(a * -0.3333333333333333) / b)
	tmp = 0.0
	if (t_1 <= -2e+18)
		tmp = t_2;
	elseif (t_1 <= 5e-109)
		tmp = Float64(sqrt(x) * Float64(2.0 - Float64(y * y)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = (a * -0.3333333333333333) / b;
	tmp = 0.0;
	if (t_1 <= -2e+18)
		tmp = t_2;
	elseif (t_1 <= 5e-109)
		tmp = sqrt(x) * (2.0 - (y * y));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+18], t$95$2, If[LessEqual[t$95$1, 5e-109], N[(N[Sqrt[x], $MachinePrecision] * N[(2.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2e18 or 5.0000000000000002e-109 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 79.4%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 90.7

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Simplified 90.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b} \]

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

         \[\leadsto \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b} \]

      8. lower-*.f64 79.5

         \[\leadsto \frac{\color{blue}{a \cdot -0.3333333333333333}}{b} \]

   8. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{a \cdot -0.3333333333333333}{b}} \]

   if -2e18 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.0000000000000002e-109

   1. Initial program 61.6%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 61.3

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Simplified 61.3%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b}\right) \]

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

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b}\right) \]

      15. lower-*.f64 61.3

          \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot -0.3333333333333333}}{b}\right) \]

   8. Simplified 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot -0.3333333333333333}{b}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{x} \cdot {y}^{2}\right) + \left(\frac{-1}{3} \cdot \frac{a}{b} + 2 \cdot \sqrt{x}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto -1 \cdot \left(\sqrt{x} \cdot {y}^{2}\right) + \color{blue}{\left(2 \cdot \sqrt{x} + \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

       2. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{x} \cdot {y}^{2}\right) + 2 \cdot \sqrt{x}\right) + \frac{-1}{3} \cdot \frac{a}{b}} \]

       3. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{x} \cdot {y}^{2}\right)\right)} + 2 \cdot \sqrt{x}\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

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

          \[\leadsto \left(\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left({y}^{2}\right)\right)} + 2 \cdot \sqrt{x}\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

       5. *-commutative N/A

          \[\leadsto \left(\sqrt{x} \cdot \left(\mathsf{neg}\left({y}^{2}\right)\right) + \color{blue}{\sqrt{x} \cdot 2}\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

       6. distribute-lft-out N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\left(\mathsf{neg}\left({y}^{2}\right)\right) + 2\right)} + \frac{-1}{3} \cdot \frac{a}{b} \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \left(\mathsf{neg}\left({y}^{2}\right)\right) + 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, \left(\mathsf{neg}\left({y}^{2}\right)\right) + 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x}, \left(\mathsf{neg}\left(\color{blue}{y \cdot y}\right)\right) + 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

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

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \color{blue}{y \cdot \left(\mathsf{neg}\left(y\right)\right)} + 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, y \cdot \color{blue}{\left(-1 \cdot y\right)} + 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \color{blue}{\mathsf{fma}\left(y, -1 \cdot y, 2\right)}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(y\right)}, 2\right), \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       14. lower-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(y\right)}, 2\right), \frac{-1}{3} \cdot \frac{a}{b}\right) \]

       15. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, \mathsf{neg}\left(y\right), 2\right), \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

       16. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, \mathsf{neg}\left(y\right), 2\right), \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

       17. lower-*.f64 31.2

           \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, -y, 2\right), \frac{\color{blue}{-0.3333333333333333 \cdot a}}{b}\right) \]

   11. Simplified 31.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(y, -y, 2\right), \frac{-0.3333333333333333 \cdot a}{b}\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\sqrt{x} \cdot \left(2 + -1 \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{\left(2 - {y}^{2}\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{\left(2 - {y}^{2}\right)} \]

       6. unpow2 N/A

          \[\leadsto \sqrt{x} \cdot \left(2 - \color{blue}{y \cdot y}\right) \]

       7. lower-*.f64 30.1

          \[\leadsto \sqrt{x} \cdot \left(2 - \color{blue}{y \cdot y}\right) \]

   14. Simplified 30.1%

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

Alternative 4: 65.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (- (* 2.0 (sqrt x)) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * sqrt(x)) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 * sqrt(x)) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * Math.sqrt(x)) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return (2.0 * math.sqrt(x)) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(2.0 * sqrt(x)) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 81.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Simplified 81.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

   2. lower-sqrt.f64 69.7

      \[\leadsto 2 \cdot \color{blue}{\sqrt{x}} - \frac{a}{b \cdot 3} \]

8. Simplified 69.7%

   \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
9. Add Preprocessing

Alternative 5: 65.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, 2 \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma -0.3333333333333333 (/ a b) (* 2.0 (sqrt x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(-0.3333333333333333, (a / b), (2.0 * sqrt(x)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(-0.3333333333333333, Float64(a / b), Float64(2.0 * sqrt(x)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 81.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Simplified 81.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   3. metadata-eval N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\sqrt{x}}, \cos y, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}}\right) \]

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

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b}\right) \]

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

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b}\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b}\right) \]

   15. lower-*.f64 81.4

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{\color{blue}{a \cdot -0.3333333333333333}}{b}\right) \]

8. Simplified 81.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a \cdot -0.3333333333333333}{b}\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b} + 2 \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{a}{b}, 2 \cdot \sqrt{x}\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{a}{b}}, 2 \cdot \sqrt{x}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \frac{a}{b}, \color{blue}{2 \cdot \sqrt{x}}\right) \]

    4. lower-sqrt.f64 69.7

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, 2 \cdot \color{blue}{\sqrt{x}}\right) \]

11. Simplified 69.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, 2 \cdot \sqrt{x}\right)} \]
12. Add Preprocessing

Alternative 6: 50.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot -0.3333333333333333}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (* a -0.3333333333333333) b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (a * -0.3333333333333333) / b;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * (-0.3333333333333333d0)) / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (a * -0.3333333333333333) / b;
}

Python

def code(x, y, z, t, a, b):
	return (a * -0.3333333333333333) / b

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(a * -0.3333333333333333) / b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (a * -0.3333333333333333) / b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 81.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Simplified 81.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

6. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot a}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}{b}} \]

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

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}}{b} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)}{b} \]

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

      \[\leadsto \frac{\color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{b} \]

   7. metadata-eval N/A

      \[\leadsto \frac{a \cdot \color{blue}{\frac{-1}{3}}}{b} \]

   8. lower-*.f64 55.6

      \[\leadsto \frac{\color{blue}{a \cdot -0.3333333333333333}}{b} \]

8. Simplified 55.6%

   \[\leadsto \color{blue}{\frac{a \cdot -0.3333333333333333}{b}} \]
9. Add Preprocessing

Alternative 7: 50.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* -0.3333333333333333 (/ a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-0.3333333333333333d0) * (a / b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / b);
}

Python

def code(x, y, z, t, a, b):
	return -0.3333333333333333 * (a / b)

Julia

function code(x, y, z, t, a, b)
	return Float64(-0.3333333333333333 * Float64(a / b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = -0.3333333333333333 * (a / b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 81.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Simplified 81.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(b \cdot \cos y\right) \cdot \sqrt{x}\right) - \frac{1}{3} \cdot a}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(b \cdot \cos y\right) \cdot \sqrt{x}\right) - \frac{1}{3} \cdot a}{b}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \left(\left(b \cdot \cos y\right) \cdot \sqrt{x}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a}}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \left(b \cdot \cos y\right) \cdot \sqrt{x}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}}{b} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(2, \color{blue}{\sqrt{x} \cdot \left(b \cdot \cos y\right)}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(2, \color{blue}{\sqrt{x} \cdot \left(b \cdot \cos y\right)}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}{b} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(2, \color{blue}{\sqrt{x}} \cdot \left(b \cdot \cos y\right), \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \color{blue}{\left(b \cdot \cos y\right)}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}{b} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \color{blue}{\cos y}\right), \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot a\right)}{b} \]

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

      \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), \color{blue}{\mathsf{neg}\left(\frac{1}{3} \cdot a\right)}\right)}{b} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), \mathsf{neg}\left(\color{blue}{a \cdot \frac{1}{3}}\right)\right)}{b} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)}{b} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), a \cdot \color{blue}{\frac{-1}{3}}\right)}{b} \]

   13. lower-*.f64 77.9

       \[\leadsto \frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), \color{blue}{a \cdot -0.3333333333333333}\right)}{b} \]

8. Simplified 77.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, \sqrt{x} \cdot \left(b \cdot \cos y\right), a \cdot -0.3333333333333333\right)}{b}} \]

9. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]

    2. lower-/.f64 55.5

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

11. Simplified 55.5%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
12. Add Preprocessing

Developer Target 1: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))