Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 98.6%

Time: 16.8s

Alternatives: 3

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: 99.1% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \cos^{-1} \left(\frac{\frac{1 + {v\_m}^{2} \cdot -5}{v\_m + -1}}{1 + v\_m}\right) \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (/ (/ (+ 1.0 (* (pow v_m 2.0) -5.0)) (+ v_m -1.0)) (+ 1.0 v_m))))

C

v_m = fabs(v);
double code(double v_m) {
	return acos((((1.0 + (pow(v_m, 2.0) * -5.0)) / (v_m + -1.0)) / (1.0 + v_m)));
}

Fortran

v_m = abs(v)
real(8) function code(v_m)
    real(8), intent (in) :: v_m
    code = acos((((1.0d0 + ((v_m ** 2.0d0) * (-5.0d0))) / (v_m + (-1.0d0))) / (1.0d0 + v_m)))
end function

Java

v_m = Math.abs(v);
public static double code(double v_m) {
	return Math.acos((((1.0 + (Math.pow(v_m, 2.0) * -5.0)) / (v_m + -1.0)) / (1.0 + v_m)));
}

Python

v_m = math.fabs(v)
def code(v_m):
	return math.acos((((1.0 + (math.pow(v_m, 2.0) * -5.0)) / (v_m + -1.0)) / (1.0 + v_m)))

Julia

v_m = abs(v)
function code(v_m)
	return acos(Float64(Float64(Float64(1.0 + Float64((v_m ^ 2.0) * -5.0)) / Float64(v_m + -1.0)) / Float64(1.0 + v_m)))
end

MATLAB

v_m = abs(v);
function tmp = code(v_m)
	tmp = acos((((1.0 + ((v_m ^ 2.0) * -5.0)) / (v_m + -1.0)) / (1.0 + v_m)));
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(N[(1.0 + N[(N[Power[v$95$m, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + v$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. difference-of-sqr-1 99.4%

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

   3. times-frac 99.4%

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

   4. +-commutative 99.4%

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

   5. pow2 99.4%

      \[\leadsto \cos^{-1} \left(\frac{1}{1 + v} \cdot \frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v - 1}\right) \]

   6. sub-neg 99.4%

      \[\leadsto \cos^{-1} \left(\frac{1}{1 + v} \cdot \frac{1 - 5 \cdot {v}^{2}}{\color{blue}{v + \left(-1\right)}}\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \cos^{-1} \left(\frac{1}{1 + v} \cdot \frac{1 - 5 \cdot {v}^{2}}{v + \color{blue}{-1}}\right) \]

4. Applied egg-rr 99.4%

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

   1. associate-*l/ 99.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 \cdot \frac{1 - 5 \cdot {v}^{2}}{v + -1}}{1 + v}\right)} \]

   2. *-un-lft-identity 99.4%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{1 - 5 \cdot {v}^{2}}{v + -1}}}{1 + v}\right) \]

   3. sub-neg 99.4%

      \[\leadsto \cos^{-1} \left(\frac{\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{v + -1}}{1 + v}\right) \]

   4. *-commutative 99.4%

      \[\leadsto \cos^{-1} \left(\frac{\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{v + -1}}{1 + v}\right) \]

   5. distribute-rgt-neg-in 99.4%

      \[\leadsto \cos^{-1} \left(\frac{\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{v + -1}}{1 + v}\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \cos^{-1} \left(\frac{\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{v + -1}}{1 + v}\right) \]

6. Applied egg-rr 99.4%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v_m v_m))) (- (* v_m v_m) 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v$95$m * v$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v$95$m * v$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \cos^{-1} -1 \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos -1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

v_m = abs(v)
function code(v_m)
	return acos(-1.0)
end

MATLAB

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0 98.6%

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

Reproduce

herbie shell --seed 2024050 -o generate:simplify
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))