ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.6%

Time: 5.8s

Alternatives: 5

Speedup: 1.0×

• 28.6% 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: 82.1% 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

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

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 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 * 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 * 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 * 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 * 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(Float64(a_m * 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 * 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[Sqrt[N[(a$95$m * b$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in a around 0 80.1%

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

   1. unpow2 80.1%

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

   2. unpow2 80.1%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

5. Simplified 99.7%

   \[\leadsto -\color{blue}{{\left(a \cdot b\right)}^{2}} \]
6. 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 54.4%

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

   3. associate-*r* 54.5%

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

   4. pow1 54.5%

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

   5. pow1/2 54.5%

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

   6. pow-prod-up 54.5%

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

   7. metadata-eval 54.5%

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

7. Applied egg-rr 54.5%

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

8. Final simplification 54.5%

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

Alternative 2: 99.7% 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 84.2%

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

3. Taylor expanded in a around 0 80.1%

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

   1. unpow2 80.1%

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

   2. unpow2 80.1%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

5. Simplified 99.7%

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

   1. unpow2 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = b_m * (a_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[(b$95$m * N[(a$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. distribute-rgt-neg-in 84.2%

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

   2. associate-*l* 92.9%

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

3. Simplified 92.9%

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

   1. neg-sub0 92.9%

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

   2. sub-neg 92.9%

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

   3. add-sqr-sqrt 44.3%

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

   4. sqrt-unprod 54.2%

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

   5. sqr-neg 54.2%

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

   6. sqrt-unprod 14.9%

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

   7. add-sqr-sqrt 26.2%

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

6. Applied egg-rr 26.2%

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

   1. +-lft-identity 26.2%

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

8. Simplified 26.2%

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

9. Final simplification 26.2%

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

Alternative 4: 28.2% accurate, 1.1× speedup

Math

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

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 * 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 84.2%

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

   1. add-sqr-sqrt 25.3%

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

   2. sqrt-unprod 26.2%

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

   3. sqr-neg 26.2%

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

   4. sqrt-unprod 26.2%

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

   5. add-sqr-sqrt 26.2%

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

   6. associate-*l* 25.9%

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

   7. swap-sqr 26.1%

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

4. Applied egg-rr 26.1%

   \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
5. Add Preprocessing

Alternative 5: 28.3% accurate, 1.1× speedup

Math

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

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(a_m * Float64(b_m * Float64(a_m * 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[(a$95$m * N[(b$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 80.1%

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

   2. associate-*r* 84.8%

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

   3. *-commutative 84.8%

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

   4. distribute-rgt-neg-in 84.8%

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

   5. distribute-rgt-neg-in 84.8%

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

   6. associate-*r* 96.2%

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

3. Simplified 96.2%

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

   1. neg-sub0 96.2%

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

   2. sub-neg 96.2%

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

   3. add-sqr-sqrt 47.6%

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

   4. sqrt-unprod 55.3%

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

   5. sqr-neg 55.3%

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

   6. sqrt-prod 14.8%

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

   7. add-sqr-sqrt 26.1%

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

6. Applied egg-rr 26.1%

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

   1. +-lft-identity 26.1%

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

8. Simplified 26.1%

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

9. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))