Radioactive exchange between two surfaces

Percentage Accurate: 86.2% → 93.1%

Time: 9.6s

Alternatives: 4

Speedup: 0.7×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

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

Alternative 1: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{\left(\sqrt[3]{y}\right)}^{12}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (pow y 4.0) 2e+307)
   (- (pow x 4.0) (pow y 4.0))
   (- (pow (cbrt y) 12.0))))

C

double code(double x, double y) {
	double tmp;
	if (pow(y, 4.0) <= 2e+307) {
		tmp = pow(x, 4.0) - pow(y, 4.0);
	} else {
		tmp = -pow(cbrt(y), 12.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.pow(y, 4.0) <= 2e+307) {
		tmp = Math.pow(x, 4.0) - Math.pow(y, 4.0);
	} else {
		tmp = -Math.pow(Math.cbrt(y), 12.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y ^ 4.0) <= 2e+307)
		tmp = Float64((x ^ 4.0) - (y ^ 4.0));
	else
		tmp = Float64(-(cbrt(y) ^ 12.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Power[y, 4.0], $MachinePrecision], 2e+307], N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], (-N[Power[N[Power[y, 1/3], $MachinePrecision], 12.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y #s(literal 4 binary64)) < 1.99999999999999997e307

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   if 1.99999999999999997e307 < (pow.f64 y #s(literal 4 binary64))

   1. Initial program 59.8%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.7%

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. neg-mul-1 81.7%

         \[\leadsto \color{blue}{-{y}^{4}} \]

   5. Simplified 81.7%

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

      1. metadata-eval 81.7%

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

      2. pow-pow 81.7%

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

      3. pow2 81.7%

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

      4. add-cube-cbrt 81.7%

         \[\leadsto -{\left(y \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right)}^{2} \]

      5. associate-*r* 81.7%

         \[\leadsto -{\color{blue}{\left(\left(y \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right)}}^{2} \]

      6. unpow-prod-down 81.7%

         \[\leadsto -\color{blue}{{\left(y \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2}} \]

      7. add-cube-cbrt 81.7%

         \[\leadsto -{\left(\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2} \]

      8. pow3 81.7%

         \[\leadsto -{\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{3}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2} \]

      9. pow2 81.7%

         \[\leadsto -{\left({\left(\sqrt[3]{y}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2} \]

      10. pow-prod-up 81.7%

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

      11. metadata-eval 81.7%

          \[\leadsto -{\left({\left(\sqrt[3]{y}\right)}^{\color{blue}{5}}\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2} \]

   7. Applied egg-rr 81.7%

      \[\leadsto -\color{blue}{{\left({\left(\sqrt[3]{y}\right)}^{5}\right)}^{2} \cdot {\left(\sqrt[3]{y}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 81.7%

         \[\leadsto -{\left({\left(\sqrt[3]{y}\right)}^{5}\right)}^{2} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} \]

      2. associate-*r* 81.7%

         \[\leadsto -\color{blue}{\left({\left({\left(\sqrt[3]{y}\right)}^{5}\right)}^{2} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \]

      3. unpow2 81.7%

         \[\leadsto -\left(\color{blue}{\left({\left(\sqrt[3]{y}\right)}^{5} \cdot {\left(\sqrt[3]{y}\right)}^{5}\right)} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} \]

      4. pow-sqr 81.7%

         \[\leadsto -\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(2 \cdot 5\right)}} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} \]

      5. metadata-eval 81.7%

         \[\leadsto -\left({\left(\sqrt[3]{y}\right)}^{\color{blue}{10}} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} \]

      6. pow-plus 81.7%

         \[\leadsto -\color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(10 + 1\right)}} \cdot \sqrt[3]{y} \]

      7. pow-plus 81.7%

         \[\leadsto -\color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(\left(10 + 1\right) + 1\right)}} \]

      8. metadata-eval 81.7%

         \[\leadsto -{\left(\sqrt[3]{y}\right)}^{\left(\color{blue}{11} + 1\right)} \]

      9. metadata-eval 81.7%

         \[\leadsto -{\left(\sqrt[3]{y}\right)}^{\color{blue}{12}} \]

   9. Simplified 81.7%

      \[\leadsto -\color{blue}{{\left(\sqrt[3]{y}\right)}^{12}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 1.12 \cdot 10^{+45}:\\ \;\;\;\;-{y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (pow x 4.0) 1.12e+45) (- (pow y 4.0)) (pow x 4.0)))

C

double code(double x, double y) {
	double tmp;
	if (pow(x, 4.0) <= 1.12e+45) {
		tmp = -pow(y, 4.0);
	} else {
		tmp = pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x ** 4.0d0) <= 1.12d+45) then
        tmp = -(y ** 4.0d0)
    else
        tmp = x ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.pow(x, 4.0) <= 1.12e+45) {
		tmp = -Math.pow(y, 4.0);
	} else {
		tmp = Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.pow(x, 4.0) <= 1.12e+45:
		tmp = -math.pow(y, 4.0)
	else:
		tmp = math.pow(x, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x ^ 4.0) <= 1.12e+45)
		tmp = Float64(-(y ^ 4.0));
	else
		tmp = x ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x ^ 4.0) <= 1.12e+45)
		tmp = -(y ^ 4.0);
	else
		tmp = x ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Power[x, 4.0], $MachinePrecision], 1.12e+45], (-N[Power[y, 4.0], $MachinePrecision]), N[Power[x, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 x #s(literal 4 binary64)) < 1.12e45

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.3%

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. neg-mul-1 95.3%

         \[\leadsto \color{blue}{-{y}^{4}} \]

   5. Simplified 95.3%

      \[\leadsto \color{blue}{-{y}^{4}} \]

   if 1.12e45 < (pow.f64 x #s(literal 4 binary64))

   1. Initial program 71.3%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

      \[\leadsto \color{blue}{{x}^{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (pow x 4.0))

C

double code(double x, double y) {
	return pow(x, 4.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x ** 4.0d0
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0)

Julia

function code(x, y)
	return x ^ 4.0
end

MATLAB

function tmp = code(x, y)
	tmp = x ^ 4.0;
end

Wolfram

code[x_, y_] := N[Power[x, 4.0], $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 56.6%

   \[\leadsto \color{blue}{{x}^{4}} \]
4. Add Preprocessing

Alternative 4: 15.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 87.1%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 62.5%

   \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
4. Step-by-step derivation

   1. neg-mul-1 62.5%

      \[\leadsto \color{blue}{-{y}^{4}} \]

5. Simplified 62.5%

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

   1. add-sqr-sqrt 18.4%

      \[\leadsto \color{blue}{\sqrt{-{y}^{4}} \cdot \sqrt{-{y}^{4}}} \]

   2. sqrt-unprod 27.3%

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

   3. sqr-neg 27.3%

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

   4. sqrt-unprod 25.0%

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

   5. add-log-exp 29.1%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{{y}^{4}} \cdot \sqrt{{y}^{4}}}\right)} \]

   6. add-sqr-sqrt 29.1%

      \[\leadsto \log \left(e^{\color{blue}{{y}^{4}}}\right) \]

   7. add-sqr-sqrt 29.1%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{{y}^{4}}} \cdot \sqrt{e^{{y}^{4}}}\right)} \]

   8. sqrt-unprod 29.1%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{{y}^{4}} \cdot e^{{y}^{4}}}\right)} \]

   9. *-un-lft-identity 29.1%

      \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot e^{\color{blue}{1 \cdot {y}^{4}}}}\right) \]

   10. exp-prod 29.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{{\left(e^{1}\right)}^{\left({y}^{4}\right)}}}\right) \]

   11. add-sqr-sqrt 29.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{{y}^{4}} \cdot \sqrt{{y}^{4}}\right)}}}\right) \]

   12. sqrt-unprod 29.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{{y}^{4} \cdot {y}^{4}}\right)}}}\right) \]

   13. sqr-neg 29.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\left(\sqrt{\color{blue}{\left(-{y}^{4}\right) \cdot \left(-{y}^{4}\right)}}\right)}}\right) \]

   14. sqrt-unprod 18.4%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{-{y}^{4}} \cdot \sqrt{-{y}^{4}}\right)}}}\right) \]

   15. add-sqr-sqrt 18.8%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(-{y}^{4}\right)}}}\right) \]

   16. exp-prod 18.8%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{e^{1 \cdot \left(-{y}^{4}\right)}}}\right) \]

   17. *-un-lft-identity 18.8%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot e^{\color{blue}{-{y}^{4}}}}\right) \]

   18. exp-neg 18.8%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{\frac{1}{e^{{y}^{4}}}}}\right) \]

   19. rgt-mult-inverse 19.7%

       \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]

7. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))