Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 22.1s

Alternatives: 4

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - \log \left(1 + \mathsf{expm1}\left(5 \cdot {v}^{2}\right)\right)}{v \cdot v + -1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (log (+ 1.0 (expm1 (* 5.0 (pow v 2.0)))))) (+ (* v v) -1.0))))

C

double code(double v) {
	return acos(((1.0 - log((1.0 + expm1((5.0 * pow(v, 2.0)))))) / ((v * v) + -1.0)));
}

Java

public static double code(double v) {
	return Math.acos(((1.0 - Math.log((1.0 + Math.expm1((5.0 * Math.pow(v, 2.0)))))) / ((v * v) + -1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - math.log((1.0 + math.expm1((5.0 * math.pow(v, 2.0)))))) / ((v * v) + -1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - log(Float64(1.0 + expm1(Float64(5.0 * (v ^ 2.0)))))) / Float64(Float64(v * v) + -1.0)))
end

Wolfram

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

   1. log1p-expm1-u 99.4%

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

   2. log1p-udef 99.4%

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

   3. pow2 99.4%

      \[\leadsto \cos^{-1} \left(\frac{1 - \log \left(1 + \mathsf{expm1}\left(5 \cdot \color{blue}{{v}^{2}}\right)\right)}{v \cdot v - 1}\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{\log \left(1 + \mathsf{expm1}\left(5 \cdot {v}^{2}\right)\right)}}{v \cdot v - 1}\right) \]

5. Final simplification 99.4%

   \[\leadsto \cos^{-1} \left(\frac{1 - \log \left(1 + \mathsf{expm1}\left(5 \cdot {v}^{2}\right)\right)}{v \cdot v + -1}\right) \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (/ (+ 1.0 (* (pow v 2.0) -5.0)) (+ v -1.0)) (+ 1.0 v))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[ArcCos[N[(N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + v), $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/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. cancel-sign-sub-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 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]

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. Final simplification 99.4%

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

Alternative 4: 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 99.4%

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

3. Taylor expanded in v around 0 98.7%

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

4. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024029 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))