Average Error: 0.5 → 0.5
Time: 15.5s
Precision: binary64
Cost: 32832
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[{\left(\frac{1}{\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, 5, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{-1} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (pow (/ 1.0 (- PI (acos (/ (fma (* v v) 5.0 -1.0) (fma v v -1.0))))) -1.0))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return pow((1.0 / (((double) M_PI) - acos((fma((v * v), 5.0, -1.0) / fma(v, v, -1.0))))), -1.0);
}

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

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

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(\frac{1}{\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, 5, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{-1}

Error

Derivation

  1. Initial program 0.5

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

  2. Applied egg-rr62.6

    \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.5

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

  5. Taylor expanded in v around 0 0.5

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

  6. Simplified0.5

    \[\leadsto {\left(\frac{1}{\pi - \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, 5, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{-1} \]
    Proof
    (acos.f64 (/.f64 (fma.f64 (*.f64 v v) 5 -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 v 2)) 5 -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 v 2) 5) -1)) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (*.f64 (pow.f64 v 2) (Rewrite<= metadata-eval (*.f64 -5 -1))) -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 v 2) -5) -1)) -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (*.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 v v)) -5) -1) -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 v (*.f64 v -5))) -1) -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (*.f64 (*.f64 v (Rewrite<= *-commutative_binary64 (*.f64 -5 v))) -1) -1) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (+.f64 (*.f64 (*.f64 v (*.f64 -5 v)) -1) (Rewrite<= metadata-eval (*.f64 1 -1))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1 (+.f64 (*.f64 v (*.f64 -5 v)) 1))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (/.f64 (*.f64 -1 (Rewrite<= fma-udef_binary64 (fma.f64 v (*.f64 -5 v) 1))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (fma.f64 v (*.f64 -5 v) 1) (fma.f64 v v -1))))): 0 points increase in error, 0 points decrease in error

  7. Final simplification0.5

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

Alternatives

Alternative 1
Error0.5
Cost26368
\[{\left(\frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{-1} \]
Alternative 2
Error0.5
Cost20032
\[\pi - \cos^{-1} \left(\frac{1 + v \cdot \left(v \cdot -5\right)}{-\mathsf{fma}\left(v, v, -1\right)}\right) \]
Alternative 3
Error0.5
Cost7232
\[\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \]
Alternative 4
Error1.3
Cost6464
\[\cos^{-1} -1 \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))