ab-angle->ABCF D

Percentage Accurate: 82.4% → 99.6%

Time: 5.3s

Alternatives: 2

Speedup: 1.0×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* (* a a) b) b)))

C

double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(((a * a) * b) * b)
end function

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

function code(a, b)
	return Float64(-Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b)
	tmp = -(((a * a) * b) * b);
end

Wolfram

code[a_, b_] := (-N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision])

Initial Program: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* (* a a) b) b)))

C

double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(((a * a) * b) * b)
end function

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

function code(a, b)
	return Float64(-Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b)
	tmp = -(((a * a) * b) * b);
end

Wolfram

code[a_, b_] := (-N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision])

Alternative 1: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \sqrt{a \cdot b} \cdot \left(-{\left(a \cdot b\right)}^{1.5}\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b) :precision binary64 (* (sqrt (* a b)) (- (pow (* a b) 1.5))))

C

a = abs(a);
b = abs(b);
double code(double a, double b) {
	return sqrt((a * b)) * -pow((a * b), 1.5);
}

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((a * b)) * -((a * b) ** 1.5d0)
end function

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b) {
	return Math.sqrt((a * b)) * -Math.pow((a * b), 1.5);
}

Python

a = abs(a)
b = abs(b)
def code(a, b):
	return math.sqrt((a * b)) * -math.pow((a * b), 1.5)

Julia

a = abs(a)
b = abs(b)
function code(a, b)
	return Float64(sqrt(Float64(a * b)) * Float64(-(Float64(a * b) ^ 1.5)))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b)
	tmp = sqrt((a * b)) * -((a * b) ^ 1.5);
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_] := N[(N[Sqrt[N[(a * b), $MachinePrecision]], $MachinePrecision] * (-N[Power[N[(a * b), $MachinePrecision], 1.5], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 81.6%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Step-by-step derivation

   1. distribute-rgt-neg-in 81.6%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)} \]

   2. associate-*l* 74.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot \left(-b\right)\right)} \]

3. Simplified 74.5%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot \left(-b\right)\right)} \]

4. Taylor expanded in a around 0 74.5%

   \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 74.5%

      \[\leadsto \color{blue}{-{a}^{2} \cdot {b}^{2}} \]

   2. unpow2 74.5%

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

   3. unpow2 74.5%

      \[\leadsto -\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

   4. swap-sqr 99.7%

      \[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   5. unpow2 99.7%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{-{\left(a \cdot b\right)}^{2}} \]
7. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   2. add-sqr-sqrt 53.7%

      \[\leadsto -\left(a \cdot b\right) \cdot \color{blue}{\left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)} \]

   3. associate-*r* 53.8%

      \[\leadsto -\color{blue}{\left(\left(a \cdot b\right) \cdot \sqrt{a \cdot b}\right) \cdot \sqrt{a \cdot b}} \]

8. Applied egg-rr 53.8%

   \[\leadsto -\color{blue}{\left(\left(a \cdot b\right) \cdot \sqrt{a \cdot b}\right) \cdot \sqrt{a \cdot b}} \]
9. Step-by-step derivation

   1. add-log-exp 37.8%

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

   2. *-un-lft-identity 37.8%

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

   3. log-prod 37.8%

      \[\leadsto -\color{blue}{\left(\log 1 + \log \left(e^{\left(a \cdot b\right) \cdot \sqrt{a \cdot b}}\right)\right)} \cdot \sqrt{a \cdot b} \]

   4. metadata-eval 37.8%

      \[\leadsto -\left(\color{blue}{0} + \log \left(e^{\left(a \cdot b\right) \cdot \sqrt{a \cdot b}}\right)\right) \cdot \sqrt{a \cdot b} \]

   5. add-log-exp 53.8%

      \[\leadsto -\left(0 + \color{blue}{\left(a \cdot b\right) \cdot \sqrt{a \cdot b}}\right) \cdot \sqrt{a \cdot b} \]

   6. *-commutative 53.8%

      \[\leadsto -\left(0 + \color{blue}{\sqrt{a \cdot b} \cdot \left(a \cdot b\right)}\right) \cdot \sqrt{a \cdot b} \]

   7. pow1/2 53.8%

      \[\leadsto -\left(0 + \color{blue}{{\left(a \cdot b\right)}^{0.5}} \cdot \left(a \cdot b\right)\right) \cdot \sqrt{a \cdot b} \]

   8. pow-plus 53.8%

      \[\leadsto -\left(0 + \color{blue}{{\left(a \cdot b\right)}^{\left(0.5 + 1\right)}}\right) \cdot \sqrt{a \cdot b} \]

   9. metadata-eval 53.8%

      \[\leadsto -\left(0 + {\left(a \cdot b\right)}^{\color{blue}{1.5}}\right) \cdot \sqrt{a \cdot b} \]

10. Applied egg-rr 53.8%

    \[\leadsto -\color{blue}{\left(0 + {\left(a \cdot b\right)}^{1.5}\right)} \cdot \sqrt{a \cdot b} \]
11. Step-by-step derivation

    1. +-lft-identity 53.8%

       \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{1.5}} \cdot \sqrt{a \cdot b} \]

12. Simplified 53.8%

    \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{1.5}} \cdot \sqrt{a \cdot b} \]

13. Final simplification 53.8%

    \[\leadsto \sqrt{a \cdot b} \cdot \left(-{\left(a \cdot b\right)}^{1.5}\right) \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \left(a \cdot b\right) \cdot \left(a \cdot \left(-b\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b) :precision binary64 (* (* a b) (* a (- b))))

C

a = abs(a);
b = abs(b);
double code(double a, double b) {
	return (a * b) * (a * -b);
}

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * b) * (a * -b)
end function

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b) {
	return (a * b) * (a * -b);
}

Python

a = abs(a)
b = abs(b)
def code(a, b):
	return (a * b) * (a * -b)

Julia

a = abs(a)
b = abs(b)
function code(a, b)
	return Float64(Float64(a * b) * Float64(a * Float64(-b)))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b)
	tmp = (a * b) * (a * -b);
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_] := N[(N[(a * b), $MachinePrecision] * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.6%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Step-by-step derivation

   1. distribute-rgt-neg-in 81.6%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)} \]

   2. associate-*l* 74.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot \left(-b\right)\right)} \]

3. Simplified 74.5%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot \left(-b\right)\right)} \]

4. Taylor expanded in a around 0 74.5%

   \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 74.5%

      \[\leadsto \color{blue}{-{a}^{2} \cdot {b}^{2}} \]

   2. unpow2 74.5%

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

   3. unpow2 74.5%

      \[\leadsto -\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

   4. swap-sqr 99.7%

      \[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   5. unpow2 99.7%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{-{\left(a \cdot b\right)}^{2}} \]
7. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

8. Applied egg-rr 99.7%

   \[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

9. Final simplification 99.7%

   \[\leadsto \left(a \cdot b\right) \cdot \left(a \cdot \left(-b\right)\right) \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))