Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 16.3s

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: 99.1% 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: 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]

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - \sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (- (/ PI 2.0) (asin (+ -1.0 (* (* v v) 4.0)))))

C

double code(double v) {
	return (((double) M_PI) / 2.0) - asin((-1.0 + ((v * v) * 4.0)));
}

Java

public static double code(double v) {
	return (Math.PI / 2.0) - Math.asin((-1.0 + ((v * v) * 4.0)));
}

Python

def code(v):
	return (math.pi / 2.0) - math.asin((-1.0 + ((v * v) * 4.0)))

Julia

function code(v)
	return Float64(Float64(pi / 2.0) - asin(Float64(-1.0 + Float64(Float64(v * v) * 4.0))))
end

MATLAB

function tmp = code(v)
	tmp = (pi / 2.0) - asin((-1.0 + ((v * v) * 4.0)));
end

Wolfram

code[v_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(4 \cdot {v}^{2} - 1\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + -1\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(-1 + 4 \cdot {v}^{2}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(4 \cdot {v}^{2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left({v}^{2} \cdot 4\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({v}^{2}\right), 4\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(v \cdot v\right), 4\right)\right)\right) \]

   8. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), 4\right)\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \cos^{-1} \color{blue}{\left(-1 + \left(v \cdot v\right) \cdot 4\right)} \]
6. Step-by-step derivation

   1. acos-asin N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right)} \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \color{blue}{\left(-1 + \left(v \cdot v\right) \cdot 4\right)}\right) \]

   4. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\color{blue}{-1} + \left(v \cdot v\right) \cdot 4\right)\right) \]

   5. asin-lowering-asin.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(-1 + \left(v \cdot v\right) \cdot 4\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(-1, \left(\left(v \cdot v\right) \cdot 4\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(v \cdot v\right), 4\right)\right)\right)\right) \]

   8. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), 4\right)\right)\right)\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right)} \]
8. Add Preprocessing

Alternative 3: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (+ -1.0 (* (* v v) 4.0))))

C

double code(double v) {
	return acos((-1.0 + ((v * v) * 4.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + ((v * v) * 4.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos((-1.0 + ((v * v) * 4.0)));
}

Python

def code(v):
	return math.acos((-1.0 + ((v * v) * 4.0)))

Julia

function code(v)
	return acos(Float64(-1.0 + Float64(Float64(v * v) * 4.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos((-1.0 + ((v * v) * 4.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(4 \cdot {v}^{2} - 1\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + -1\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(-1 + 4 \cdot {v}^{2}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(4 \cdot {v}^{2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left({v}^{2} \cdot 4\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({v}^{2}\right), 4\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(v \cdot v\right), 4\right)\right)\right) \]

   8. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), 4\right)\right)\right) \]

5. Simplified 98.8%

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

Alternative 4: 97.9% 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 99.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{-1}\right) \]
4. Step-by-step derivation

5. Simplified 98.2%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))