Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 44.7s

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: 99.1% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\\ e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\frac{{\left(1 + t\_0\right)}^{2}}{2 + t\_0} - \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)\right)\right)} - 1 \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (acos (/ (- 1.0 (* 5.0 (pow v 2.0))) (- (pow v 2.0) 1.0)))))
   (-
    (exp
     (expm1
      (log1p
       (log1p
        (-
         (/ (pow (+ 1.0 t_0) 2.0) (+ 2.0 t_0))
         (/
          1.0
          (+ (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0))) 2.0)))))))
    1.0)))

C

double code(double v) {
	double t_0 = acos(((1.0 - (5.0 * pow(v, 2.0))) / (pow(v, 2.0) - 1.0)));
	return exp(expm1(log1p(log1p(((pow((1.0 + t_0), 2.0) / (2.0 + t_0)) - (1.0 / (acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0))) + 2.0))))))) - 1.0;
}

Julia

function code(v)
	t_0 = acos(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / Float64((v ^ 2.0) - 1.0)))
	return Float64(exp(expm1(log1p(log1p(Float64(Float64((Float64(1.0 + t_0) ^ 2.0) / Float64(2.0 + t_0)) - Float64(1.0 / Float64(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0))) + 2.0))))))) - 1.0)
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcCos[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[v, 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(Exp[N[Log[1 + N[Log[1 + N[(N[(N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. pow2 99.2%

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

   4. fma-neg 99.2%

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

   5. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}} - 1 \]

   2. sub-neg 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} - 1 \]

   3. *-commutative 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} - 1 \]

   4. distribute-rgt-neg-in 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} - 1 \]

   5. metadata-eval 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} - 1 \]

6. Applied egg-rr 99.2%

   \[\leadsto e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}} - 1 \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}\right)\right)\right)} - 1 \]

   2. expm1-define 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1}\right)\right)\right)} - 1 \]

   3. expm1-log1p-u 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}} - 1\right)\right)\right)} - 1 \]

   4. flip-- 99.2%

      \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{\frac{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} \cdot e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} - 1 \cdot 1}{e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} + 1}}\right)\right)\right)} - 1 \]

8. Applied egg-rr 99.2%

   \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{\frac{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2} - \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}\right)\right)\right)} - 1 \]

9. Taylor expanded in v around inf 99.2%

   \[\leadsto e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{\frac{{\left(1 + \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{2}}{2 + \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}} - \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)\right)\right)} - 1 \]
10. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1 \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (exp (log1p (acos (/ (- 1.0 (* 5.0 (pow v 2.0))) (fma v v -1.0))))) 1.0))

C

double code(double v) {
	return exp(log1p(acos(((1.0 - (5.0 * pow(v, 2.0))) / fma(v, v, -1.0))))) - 1.0;
}

Julia

function code(v)
	return Float64(exp(log1p(acos(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / fma(v, v, -1.0))))) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. pow2 99.2%

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

   4. fma-neg 99.2%

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

   5. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (/ (- 1.0 (* 5.0 (pow v 2.0))) (fma v v -1.0))))))

C

double code(double v) {
	return expm1(log1p(acos(((1.0 - (5.0 * pow(v, 2.0))) / fma(v, v, -1.0)))));
}

Julia

function code(v)
	return expm1(log1p(acos(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / fma(v, v, -1.0)))))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. pow2 99.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right)\right)\right) \]

   3. fma-neg 99.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)\right) \]

   4. metadata-eval 99.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. pow2 99.2%

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

   4. fma-neg 99.2%

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

   5. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. log1p-undefine 99.2%

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

   2. rem-exp-log 99.2%

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

   3. +-commutative 99.2%

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

   4. sub-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

Alternative 5: 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 99.2%

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

Alternative 6: 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 99.2%

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

3. Taylor expanded in v around 0 98.0%

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

Reproduce

herbie shell --seed 2024052 -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))))