ab-angle->ABCF D

Percentage Accurate: 82.2% → 99.6%

Time: 3.7s

Alternatives: 4

Speedup: 1.0×

• 27.7% 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: 82.2% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in a around 0 75.3%

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

   1. unpow2 75.3%

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

   2. unpow2 75.3%

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

   3. swap-sqr 99.6%

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

   4. unpow2 99.6%

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

4. Simplified 99.6%

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

   1. unpow2 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 75.6% accurate, 1.0× speedup

Math

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

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(a * a) * Float64(b * Float64(-b)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in a around 0 75.3%

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

   1. unpow2 75.3%

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

   2. unpow2 75.3%

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

4. Simplified 75.3%

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

5. Final simplification 75.3%

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

Alternative 3: 28.7% accurate, 1.1× speedup

Math

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

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(a * Float64(a * Float64(b * b)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. distribute-rgt-neg-in 85.5%

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

   2. associate-*l* 95.9%

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

   3. associate-*l* 93.9%

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

3. Simplified 93.9%

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

   1. expm1-log1p-u 72.8%

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

   2. expm1-udef 53.8%

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

   3. log1p-udef 53.8%

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

   4. add-exp-log 74.9%

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

   5. add-sqr-sqrt 37.7%

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

   6. sqrt-unprod 46.6%

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

   7. sqr-neg 46.6%

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

   8. sqrt-unprod 12.3%

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

   9. add-sqr-sqrt 27.8%

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

   10. associate-*l* 27.8%

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

5. Applied egg-rr 27.8%

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

   1. +-commutative 27.8%

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

   2. associate--l+ 27.6%

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

   3. metadata-eval 27.6%

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

   4. +-rgt-identity 27.6%

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

7. Simplified 27.6%

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

8. Final simplification 27.6%

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

Alternative 4: 28.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. distribute-rgt-neg-in 85.5%

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

   2. associate-*l* 95.9%

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

   3. associate-*l* 93.9%

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

3. Simplified 93.9%

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

   1. expm1-log1p-u 72.8%

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

   2. expm1-udef 53.8%

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

   3. log1p-udef 53.8%

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

   4. add-exp-log 74.9%

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

   5. add-sqr-sqrt 37.7%

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

   6. sqrt-unprod 46.6%

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

   7. sqr-neg 46.6%

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

   8. sqrt-unprod 12.3%

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

   9. add-sqr-sqrt 27.8%

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

   10. associate-*l* 27.8%

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

5. Applied egg-rr 27.8%

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

   1. +-commutative 27.8%

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

   2. associate--l+ 27.6%

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

   3. metadata-eval 27.6%

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

   4. +-rgt-identity 27.6%

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

7. Simplified 27.6%

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

8. Taylor expanded in a around 0 27.6%

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

   1. unpow2 27.6%

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

   2. associate-*r* 27.5%

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

   3. *-commutative 27.5%

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

10. Simplified 27.5%

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

11. Final simplification 27.5%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))