Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 14.4s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0))))))

C

double code(double v) {
	return expm1(log1p(acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0)))));
}

Julia

function code(v)
	return expm1(log1p(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))
end

Wolfram

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Step-by-step derivation

   1. sub-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right) \]

   2. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{1 + \left(-5 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}{v \cdot v - 1}\right) \]

   3. +-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)\right) + 1}}{v \cdot v - 1}\right) \]

   4. *-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 5}\right) + 1}{v \cdot v - 1}\right) \]

   5. distribute-rgt-neg-in 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-5\right)} + 1}{v \cdot v - 1}\right) \]

   6. fma-define 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\left(-v\right) \cdot \left(-v\right), -5, 1\right)}}{v \cdot v - 1}\right) \]

   7. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{v \cdot v - 1}\right) \]

   8. metadata-eval 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{v \cdot v - 1}\right) \]

   9. fma-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. metadata-eval 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

   2. fma-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   3. fma-undefine 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot -5 + 1}}{v \cdot v - 1}\right) \]

   4. metadata-eval 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\left(v \cdot v\right) \cdot \color{blue}{\left(-5\right)} + 1}{v \cdot v - 1}\right) \]

   5. distribute-rgt-neg-in 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-\left(v \cdot v\right) \cdot 5\right)} + 1}{v \cdot v - 1}\right) \]

   6. *-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\left(-\color{blue}{5 \cdot \left(v \cdot v\right)}\right) + 1}{v \cdot v - 1}\right) \]

   7. +-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right) \]

   8. sub-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   9. expm1-log1p-u 99.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)} \]

   10. expm1-undefine 99.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} - 1} \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-define 99.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

8. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (fma (* v v) -5.0 1.0) (fma v v -1.0))))

C

double code(double v) {
	return acos((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return acos(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Step-by-step derivation

   1. sub-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right) \]

   2. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{1 + \left(-5 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}{v \cdot v - 1}\right) \]

   3. +-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)\right) + 1}}{v \cdot v - 1}\right) \]

   4. *-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 5}\right) + 1}{v \cdot v - 1}\right) \]

   5. distribute-rgt-neg-in 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-5\right)} + 1}{v \cdot v - 1}\right) \]

   6. fma-define 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\left(-v\right) \cdot \left(-v\right), -5, 1\right)}}{v \cdot v - 1}\right) \]

   7. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{v \cdot v - 1}\right) \]

   8. metadata-eval 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{v \cdot v - 1}\right) \]

   9. fma-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{-1 + v \cdot v}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* (* v v) 5.0)) (+ -1.0 (* v v)))))

C

double code(double v) {
	return acos(((1.0 - ((v * v) * 5.0)) / (-1.0 + (v * v))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - ((v * v) * 5.0d0)) / ((-1.0d0) + (v * v))))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - ((v * v) * 5.0)) / (-1.0 + (v * v))));
}

Python

def code(v):
	return math.acos(((1.0 - ((v * v) * 5.0)) / (-1.0 + (v * v))))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / Float64(-1.0 + Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - ((v * v) * 5.0)) / (-1.0 + (v * v))));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Final simplification 99.5%

   \[\leadsto \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{-1 + v \cdot v}\right) \]
4. Add Preprocessing

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{-1}\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (/ (- 1.0 (* (* v v) 5.0)) -1.0)))

C

double code(double v) {
	return acos(((1.0 - ((v * v) * 5.0)) / -1.0));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - ((v * v) * 5.0d0)) / (-1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - ((v * v) * 5.0)) / -1.0));
}

Python

def code(v):
	return math.acos(((1.0 - ((v * v) * 5.0)) / -1.0))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / -1.0))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - ((v * v) * 5.0)) / -1.0));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0 98.7%

   \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{-1}}\right) \]

4. Final simplification 98.7%

   \[\leadsto \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{-1}\right) \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

double code(double v) {
	return acos(-1.0);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-1.0);
}

Python

def code(v):
	return math.acos(-1.0)

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-1.0);
end

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0 98.1%

   \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1}}{v \cdot v - 1}\right) \]

4. Taylor expanded in v around 0 98.6%

   \[\leadsto \cos^{-1} \left(\frac{1}{\color{blue}{-1}}\right) \]
5. Step-by-step derivation

   1. metadata-eval 98.6%

      \[\leadsto \cos^{-1} \color{blue}{-1} \]

   2. *-un-lft-identity 98.6%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} -1} \]

6. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} -1} \]
7. Step-by-step derivation

   1. *-lft-identity 98.6%

      \[\leadsto \color{blue}{\cos^{-1} -1} \]

8. Simplified 98.6%

   \[\leadsto \color{blue}{\cos^{-1} -1} \]
9. Add Preprocessing

Alternative 6: 0.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -5 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -5.0))

C

double code(double v) {
	return acos(-5.0);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-5.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-5.0);
}

Python

def code(v):
	return math.acos(-5.0)

Julia

function code(v)
	return acos(-5.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-5.0);
end

Wolfram

code[v_] := N[ArcCos[-5.0], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Step-by-step derivation

   1. sub-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right) \]

   2. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{1 + \left(-5 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}{v \cdot v - 1}\right) \]

   3. +-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)\right) + 1}}{v \cdot v - 1}\right) \]

   4. *-commutative 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 5}\right) + 1}{v \cdot v - 1}\right) \]

   5. distribute-rgt-neg-in 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-5\right)} + 1}{v \cdot v - 1}\right) \]

   6. fma-define 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\left(-v\right) \cdot \left(-v\right), -5, 1\right)}}{v \cdot v - 1}\right) \]

   7. sqr-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{v \cdot v - 1}\right) \]

   8. metadata-eval 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{v \cdot v - 1}\right) \]

   9. fma-neg 99.5%

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 0.0%

   \[\leadsto \cos^{-1} \color{blue}{-5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))