Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 93.9% → 99.9%

Time: 5.6s

Alternatives: 7

Speedup: 0.8×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))

C

double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))

C

double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+57) (fma x (* y y) x) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+57) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+57)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+57], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999972e57

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   if 4.99999999999999972e57 < (*.f64 y y)

   1. Initial program 85.4%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 85.4%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
   3. Step-by-step derivation

      1. unpow2 85.4%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (/ (/ (/ 1.0 x) (hypot 1.0 y)) (hypot 1.0 y))))

C

double code(double x, double y) {
	return 1.0 / (((1.0 / x) / hypot(1.0, y)) / hypot(1.0, y));
}

Java

public static double code(double x, double y) {
	return 1.0 / (((1.0 / x) / Math.hypot(1.0, y)) / Math.hypot(1.0, y));
}

Python

def code(x, y):
	return 1.0 / (((1.0 / x) / math.hypot(1.0, y)) / math.hypot(1.0, y))

Julia

function code(x, y)
	return Float64(1.0 / Float64(Float64(Float64(1.0 / x) / hypot(1.0, y)) / hypot(1.0, y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (((1.0 / x) / hypot(1.0, y)) / hypot(1.0, y));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(N[(1.0 / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[x \cdot \left(1 + y \cdot y\right) \]
2. Step-by-step derivation

   1. +-commutative 94.5%

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

   2. distribute-lft-in 94.5%

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

   3. *-rgt-identity 94.5%

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

3. Applied egg-rr 94.5%

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

   1. flip-+ 42.6%

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

   2. clear-num 42.5%

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

   3. clear-num 42.6%

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

   4. flip-+ 94.3%

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

   5. fma-def 94.4%

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

5. Applied egg-rr 94.4%

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

   1. fma-udef 94.3%

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

   2. *-commutative 94.3%

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

   3. distribute-lft1-in 94.3%

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

   4. /-rgt-identity 94.3%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y + 1}{1}} \cdot x}} \]

   5. associate-/r/ 94.3%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{y \cdot y + 1}{\frac{1}{x}}}}} \]

   6. clear-num 94.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{x}}{y \cdot y + 1}}} \]

   7. add-sqr-sqrt 46.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}{y \cdot y + 1}} \]

   8. add-sqr-sqrt 46.6%

      \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{\sqrt{y \cdot y + 1} \cdot \sqrt{y \cdot y + 1}}}} \]

   9. times-frac 46.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}} \cdot \frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}}}} \]

   10. inv-pow 46.6%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot y + 1}} \cdot \frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}}} \]

   11. sqrt-pow1 46.7%

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

   12. metadata-eval 46.7%

       \[\leadsto \frac{1}{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot y + 1}} \cdot \frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}}} \]

   13. +-commutative 46.7%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + y \cdot y}}} \cdot \frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}}} \]

   14. hypot-1-def 46.7%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, y\right)}} \cdot \frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot y + 1}}} \]

   15. inv-pow 46.7%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot y + 1}}} \]

   16. sqrt-pow1 46.6%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot y + 1}}} \]

   17. metadata-eval 46.6%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot y + 1}}} \]

   18. +-commutative 46.6%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + y \cdot y}}}} \]

   19. hypot-1-def 48.8%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, y\right)}}} \]

7. Applied egg-rr 48.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)}}} \]
8. Step-by-step derivation

   1. associate-*l/ 48.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{-0.5} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}}} \]

   2. associate-*r/ 48.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{\mathsf{hypot}\left(1, y\right)}}}{\mathsf{hypot}\left(1, y\right)}} \]

   3. pow-sqr 99.7%

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}} \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{1}{\frac{\frac{{x}^{\color{blue}{-1}}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}} \]

   5. unpow-1 99.7%

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{\frac{1}{x}}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}} \]

9. Simplified 99.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}}} \]

10. Final simplification 99.7%

    \[\leadsto \frac{1}{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, y\right)}}{\mathsf{hypot}\left(1, y\right)}} \]

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+57) (* x (+ 1.0 (* y y))) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+57) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d+57) then
        tmp = x * (1.0d0 + (y * y))
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+57) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+57:
		tmp = x * (1.0 + (y * y))
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+57)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+57)
		tmp = x * (1.0 + (y * y));
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+57], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999972e57

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]

   if 4.99999999999999972e57 < (*.f64 y y)

   1. Initial program 85.4%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 85.4%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
   3. Step-by-step derivation

      1. unpow2 85.4%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+57) (+ x (* x (* y y))) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+57) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d+57) then
        tmp = x + (x * (y * y))
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+57) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+57:
		tmp = x + (x * (y * y))
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+57)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+57)
		tmp = x + (x * (y * y));
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+57], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999972e57

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   if 4.99999999999999972e57 < (*.f64 y y)

   1. Initial program 85.4%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 85.4%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
   3. Step-by-step derivation

      1. unpow2 85.4%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 5: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.0) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.0) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1.0d0) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.0) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.0:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.0)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.0)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.0], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{x} \]

   if 1 < (*.f64 y y)

   1. Initial program 87.6%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 85.8%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
   3. Step-by-step derivation

      1. unpow2 85.8%

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

      2. *-commutative 85.8%

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

   4. Simplified 85.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 6: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.002) x (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.002) {
		tmp = x;
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 0.002d0) then
        tmp = x
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.002) {
		tmp = x;
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.002:
		tmp = x
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.002)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.002)
		tmp = x;
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.002], x, N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2e-3

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{x} \]

   if 2e-3 < (*.f64 y y)

   1. Initial program 87.6%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 85.8%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
   3. Step-by-step derivation

      1. unpow2 85.8%

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

      2. associate-*l* 98.0%

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

   4. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 7: 52.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 94.5%

   \[x \cdot \left(1 + y \cdot y\right) \]

2. Taylor expanded in y around 0 57.2%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 57.2%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ x (* (* x y) y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x + ((x * y) * y)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x * y) * y))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1.0 (* y y))))