Average Error: 0.0 → 0.0
Time: 5.8s
Precision: binary64
Cost: 26496
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\sqrt{e^{2 \cdot \mathsf{log1p}\left(v \cdot \left(-v\right)\right)} \cdot \left(0.125 \cdot \mathsf{fma}\left(-3, v \cdot v, 1\right)\right)} \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (sqrt (* (exp (* 2.0 (log1p (* v (- v))))) (* 0.125 (fma -3.0 (* v v) 1.0)))))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return sqrt((exp((2.0 * log1p((v * -v)))) * (0.125 * fma(-3.0, (v * v), 1.0))));
}

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end
↓
function code(v)
	return sqrt(Float64(exp(Float64(2.0 * log1p(Float64(v * Float64(-v))))) * Float64(0.125 * fma(-3.0, Float64(v * v), 1.0))))
end

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_] := N[Sqrt[N[(N[Exp[N[(2.0 * N[Log[1 + N[(v * (-v)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{e^{2 \cdot \mathsf{log1p}\left(v \cdot \left(-v\right)\right)} \cdot \left(0.125 \cdot \mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}

Error

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(1 - v \cdot v\right)\right)} \]
    Proof
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (+.f64 1 (*.f64 (*.f64 v v) -3))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (+.f64 1 (*.f64 (*.f64 v v) (Rewrite<= metadata-eval (neg.f64 3))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 3))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (+.f64 1 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 v v)))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 2) 4) (sqrt.f64 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))): 3 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{\sqrt{0.125 \cdot \left(\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot {\left(1 - v \cdot v\right)}^{2}\right)}} \]

  4. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 \cdot \mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}} \]
    Proof
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (fma.f64 -3 (*.f64 v v) 1)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (fma.f64 -3 (Rewrite<= unpow2_binary64 (pow.f64 v 2)) 1)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -3 (pow.f64 v 2)) 1))))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 v 2) -3)) 1)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (+.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 v v)) -3) 1)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 v (*.f64 v -3))) 1)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (*.f64 (pow.f64 (-.f64 1 (*.f64 v v)) 2) (*.f64 1/8 (Rewrite<= fma-udef_binary64 (fma.f64 v (*.f64 v -3) 1))))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/8 (fma.f64 v (*.f64 v -3) 1)) (pow.f64 (-.f64 1 (*.f64 v v)) 2)))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/8 (*.f64 (fma.f64 v (*.f64 v -3) 1) (pow.f64 (-.f64 1 (*.f64 v v)) 2))))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr0.0

    \[\leadsto \sqrt{\color{blue}{e^{2 \cdot \mathsf{log1p}\left(v \cdot \left(-v\right)\right)}} \cdot \left(0.125 \cdot \mathsf{fma}\left(-3, v \cdot v, 1\right)\right)} \]

  6. Final simplification0.0

    \[\leadsto \sqrt{e^{2 \cdot \mathsf{log1p}\left(v \cdot \left(-v\right)\right)} \cdot \left(0.125 \cdot \mathsf{fma}\left(-3, v \cdot v, 1\right)\right)} \]

Alternatives

Alternative 1
Error0.0
Cost13696
\[\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 + \left(v \cdot v\right) \cdot -0.375\right)} \]
Alternative 2
Error0.0
Cost7232
\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125 + \left(v \cdot v\right) \cdot -0.375} \]
Alternative 3
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]
Alternative 4
Error0.7
Cost6848
\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]
Alternative 5
Error0.7
Cost6464
\[\sqrt{0.125} \]

Error

Reproduce

herbie shell --seed 2022328 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))