Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.1%

Time: 15.4s

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: 99.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (+ (exp (log1p (acos (/ (+ 1.0 (* (* v v) -5.0)) (fma v v -1.0))))) -1.0))

C

double code(double v) {
	return exp(log1p(acos(((1.0 + ((v * v) * -5.0)) / fma(v, v, -1.0))))) + -1.0;
}

Julia

function code(v)
	return Float64(exp(log1p(acos(Float64(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / fma(v, v, -1.0))))) + -1.0)
end

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-undefine 98.8%

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

   3. sub-neg 98.8%

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

   4. *-commutative 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

   6. pow2 98.8%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2}} \cdot \left(-5\right)}{v \cdot v - 1}\right)\right)} - 1 \]

   7. metadata-eval 98.8%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{v \cdot v - 1}\right)\right)} - 1 \]

   8. fma-neg 98.8%

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

   9. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. unpow2 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

   \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} + -1 \]
8. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-undefine 98.8%

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

   3. sub-neg 98.8%

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

   4. *-commutative 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

   6. pow2 98.8%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2}} \cdot \left(-5\right)}{v \cdot v - 1}\right)\right)} - 1 \]

   7. metadata-eval 98.8%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{v \cdot v - 1}\right)\right)} - 1 \]

   8. fma-neg 98.8%

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

   9. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in v around inf 98.8%

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

   1. unpow2 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in v around 0 98.1%

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

   1. unpow2 98.8%

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

5. Applied egg-rr 98.1%

   \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

6. Final simplification 98.1%

   \[\leadsto \cos^{-1} \left(\left(v \cdot v\right) \cdot 4 + -1\right) \]
7. 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 98.8%

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

3. Taylor expanded in v around 0 98.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]

4. Taylor expanded in v around 0 97.2%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
5. 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 98.8%

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

3. Taylor expanded in v around inf 0.0%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))