ab-angle->ABCF D

Percentage Accurate: 82.7% → 99.6%

Time: 3.8s

Alternatives: 4

Speedup: 1.0×

• 28.5% 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: 82.7% 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

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (* (* b (* a (sqrt b))) (* (sqrt (* b a)) (- (sqrt a)))))

C

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

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * (a * sqrt(b))) * (sqrt((b * a)) * -sqrt(a))
end function

Java

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

Python

a = abs(a)
b = abs(b)
[a, b] = sort([a, b])
def code(a, b):
	return (b * (a * math.sqrt(b))) * (math.sqrt((b * a)) * -math.sqrt(a))

Julia

a = abs(a)
b = abs(b)
a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b * Float64(a * sqrt(b))) * Float64(sqrt(Float64(b * a)) * Float64(-sqrt(a))))
end

MATLAB

a = abs(a)
b = abs(b)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b * (a * sqrt(b))) * (sqrt((b * a)) * -sqrt(a));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(b * N[(a * N[Sqrt[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(b * a), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in a around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

4. Simplified 99.7%

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

   1. unpow2 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 95.4%

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

   4. associate-*l* 84.1%

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

   5. add-sqr-sqrt 40.5%

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

   6. swap-sqr 49.6%

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

   7. associate-*r* 51.4%

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

6. Applied egg-rr 51.4%

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

   1. add-cube-cbrt 51.1%

      \[\leadsto -\left(b \cdot \left(a \cdot \sqrt{b}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{a \cdot \sqrt{b}} \cdot \sqrt[3]{a \cdot \sqrt{b}}\right) \cdot \sqrt[3]{a \cdot \sqrt{b}}\right)} \]

   2. pow3 51.1%

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

8. Applied egg-rr 51.1%

   \[\leadsto -\left(b \cdot \left(a \cdot \sqrt{b}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{a \cdot \sqrt{b}}\right)}^{3}} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 51.4%

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

   2. *-commutative 51.4%

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

   3. add-sqr-sqrt 23.3%

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

   4. associate-*r* 23.4%

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

   5. sqrt-prod 23.4%

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

   6. *-commutative 23.4%

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

10. Applied egg-rr 23.4%

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

11. Final simplification 23.4%

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

Alternative 2: 76.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (* a a) (* b (- b))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (a * a) * (b * -b);
end

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in a around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

4. Simplified 78.5%

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

5. Final simplification 78.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (* b a) (* b (- a))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b * a) * (b * -a);
end

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in a around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

4. Simplified 99.7%

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

   1. unpow2 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 28.3% accurate, 1.1× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* a (* a (* b b))))

C

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

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * (a * (b * b))
end function

Java

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

Python

a = abs(a)
b = abs(b)
[a, b] = sort([a, b])
def code(a, b):
	return a * (a * (b * b))

Julia

a = abs(a)
b = abs(b)
a, b = sort([a, b])
function code(a, b)
	return Float64(a * Float64(a * Float64(b * b)))
end

MATLAB

a = abs(a)
b = abs(b)
a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = a * (a * (b * b));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. distribute-rgt-neg-in 84.1%

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

   2. associate-*l* 95.4%

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

   3. associate-*l* 93.5%

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

3. Simplified 93.5%

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

   1. expm1-log1p-u 73.8%

      \[\leadsto a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot b\right) \cdot \left(-b\right)\right)\right)} \]

   2. expm1-udef 59.7%

      \[\leadsto a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot b\right) \cdot \left(-b\right)\right)} - 1\right)} \]

   3. log1p-udef 59.7%

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

   4. add-exp-log 79.3%

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

   5. add-sqr-sqrt 39.9%

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

   6. sqrt-unprod 57.6%

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

   7. sqr-neg 57.6%

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

   8. sqrt-unprod 18.4%

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

   9. add-sqr-sqrt 34.5%

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

   10. associate-*l* 34.5%

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

5. Applied egg-rr 34.5%

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

   1. +-commutative 34.5%

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

   2. associate--l+ 34.3%

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

   3. metadata-eval 34.3%

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

   4. +-rgt-identity 34.3%

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

7. Simplified 34.3%

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

8. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))