ab-angle->ABCF D

Percentage Accurate: 81.7% → 99.6%

Time: 5.3s

Alternatives: 2

Speedup: 1.0×

• 27.6% 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: 81.7% 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

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

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (pow (* a_m b_m) 1.5) (* (sqrt a_m) (- (sqrt b_m)))))

C

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

Fortran

a_m = abs(a)
b_m = abs(b)
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = ((a_m * b_m) ** 1.5d0) * (sqrt(a_m) * -sqrt(b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(N[Power[N[(a$95$m * b$95$m), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[Sqrt[a$95$m], $MachinePrecision] * (-N[Sqrt[b$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

   1. add-cbrt-cube 60.4%

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

   2. pow3 60.4%

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

   3. associate-*l* 58.9%

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

   4. swap-sqr 67.9%

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

   5. pow2 67.9%

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

4. Applied egg-rr 67.9%

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

5. Taylor expanded in a around 0 48.3%

   \[\leadsto -\sqrt[3]{\color{blue}{{a}^{6} \cdot {b}^{6}}} \]
6. Step-by-step derivation

   1. metadata-eval 48.3%

      \[\leadsto -\sqrt[3]{{a}^{6} \cdot {b}^{\color{blue}{\left(2 \cdot 3\right)}}} \]

   2. pow-sqr 48.3%

      \[\leadsto -\sqrt[3]{{a}^{6} \cdot \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)}} \]

   3. metadata-eval 48.3%

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

   4. pow-sqr 48.3%

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

   5. swap-sqr 55.2%

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

   6. cube-prod 55.2%

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

   7. cube-prod 67.9%

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

   8. pow-sqr 67.9%

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

   9. metadata-eval 67.9%

      \[\leadsto -\sqrt[3]{{\left(a \cdot b\right)}^{\color{blue}{6}}} \]

7. Simplified 67.9%

   \[\leadsto -\sqrt[3]{\color{blue}{{\left(a \cdot b\right)}^{6}}} \]
8. Step-by-step derivation

   1. pow1/3 66.7%

      \[\leadsto -\color{blue}{{\left({\left(a \cdot b\right)}^{6}\right)}^{0.3333333333333333}} \]

   2. pow-pow 99.5%

      \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

   3. metadata-eval 99.5%

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

   4. pow2 99.5%

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

   5. add-sqr-sqrt 54.0%

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

   6. associate-*l* 54.0%

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

   7. *-commutative 54.0%

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

   8. sqrt-prod 29.1%

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

   9. associate-*l* 28.4%

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

   10. pow1 28.4%

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

   11. pow1/2 28.4%

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

   12. pow-prod-up 28.5%

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

   13. metadata-eval 28.5%

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

9. Applied egg-rr 28.5%

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

    1. associate-*r* 29.2%

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

    2. *-commutative 29.2%

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

11. Simplified 29.2%

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

12. Final simplification 29.2%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m) :precision binary64 (* (* a_m b_m) (* a_m (- b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return (a_m * b_m) * (a_m * -b_m);
}

Fortran

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

Java

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

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return (a_m * b_m) * (a_m * -b_m)

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(a$95$m * (-b$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in a around 0 72.6%

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

   1. unpow2 72.6%

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

   2. unpow2 72.6%

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

   3. swap-sqr 99.5%

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

   4. unpow2 99.5%

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

5. Simplified 99.5%

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

   1. unpow2 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \left(a \cdot b\right) \cdot \left(a \cdot \left(-b\right)\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))