Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%
Time: 45.1s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

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

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

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

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

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

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]

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

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

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

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

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

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

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]

\begin{array}{l}

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

Alternative 1: 99.2% accurate, 0.3× speedup?

\[\begin{array}{l} \\ e^{\log \cos^{-1} \left(\frac{v \cdot \left(v \cdot -5\right) + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)} \end{array} \]
(FPCore (v)
 :precision binary64
 (exp (log (acos (/ (+ (* v (* v -5.0)) 1.0) (fma v v -1.0))))))
double code(double v) {
	return exp(log(acos((((v * (v * -5.0)) + 1.0) / fma(v, v, -1.0)))));
}

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

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

\begin{array}{l}

\\
e^{\log \cos^{-1} \left(\frac{v \cdot \left(v \cdot -5\right) + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)}
\end{array}
Derivation

  1. Initial program 99.0%

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

    1. add-exp-log99.0%

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

    2. sub-neg99.0%

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

    3. +-commutative99.0%

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

    4. *-commutative99.0%

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

    5. distribute-rgt-neg-in99.0%

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

    6. fma-def99.0%

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

    7. metadata-eval99.0%

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

    8. fma-neg99.0%

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

    9. metadata-eval99.0%

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

  3. Applied egg-rr99.0%

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

    1. fma-udef99.0%

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

    2. associate-*l*99.0%

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

  5. Applied egg-rr99.0%

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

  6. Final simplification99.0%

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

Alternative 2: 99.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (+ -1.0 (* v v)))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / (-1.0 + (v * v))));
}

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

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

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

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

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

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

\begin{array}{l}

\\
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right)
\end{array}
Derivation

  1. Initial program 99.0%

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

  2. Final simplification99.0%

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

Alternative 3: 97.8% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]
(FPCore (v) :precision binary64 (acos -1.0))
double code(double v) {
	return acos(-1.0);
}

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

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

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

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

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

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

\begin{array}{l}

\\
\cos^{-1} -1
\end{array}
Derivation

  1. Initial program 99.0%

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

  2. Taylor expanded in v around 0 97.6%

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

  3. Final simplification97.6%

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

Reproduce

?
herbie shell --seed 2023279 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))