Average Error: 0.4 → 0.4
Time: 9.4s
Precision: binary64
Cost: 14464
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
\[\frac{1 + \left(v \cdot v\right) \cdot -5}{t \cdot \left(\pi \cdot \left(\sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))
(FPCore (v t)
 :precision binary64
 (/
  (+ 1.0 (* (* v v) -5.0))
  (* t (* PI (* (sqrt (* 2.0 (+ 1.0 (* (* v v) -3.0)))) (- 1.0 (* v v)))))))
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}
double code(double v, double t) {
	return (1.0 + ((v * v) * -5.0)) / (t * (((double) M_PI) * (sqrt((2.0 * (1.0 + ((v * v) * -3.0)))) * (1.0 - (v * v)))));
}
public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((Math.PI * t) * Math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}
public static double code(double v, double t) {
	return (1.0 + ((v * v) * -5.0)) / (t * (Math.PI * (Math.sqrt((2.0 * (1.0 + ((v * v) * -3.0)))) * (1.0 - (v * v)))));
}
def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (((math.pi * t) * math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)))
def code(v, t):
	return (1.0 + ((v * v) * -5.0)) / (t * (math.pi * (math.sqrt((2.0 * (1.0 + ((v * v) * -3.0)))) * (1.0 - (v * v)))))
function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end
function code(v, t)
	return Float64(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / Float64(t * Float64(pi * Float64(sqrt(Float64(2.0 * Float64(1.0 + Float64(Float64(v * v) * -3.0)))) * Float64(1.0 - Float64(v * v))))))
end
function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((pi * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
end
function tmp = code(v, t)
	tmp = (1.0 + ((v * v) * -5.0)) / (t * (pi * (sqrt((2.0 * (1.0 + ((v * v) * -3.0)))) * (1.0 - (v * v)))));
end
code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
code[v_, t_] := N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(t * N[(Pi * N[(N[Sqrt[N[(2.0 * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
\frac{1 + \left(v \cdot v\right) \cdot -5}{t \cdot \left(\pi \cdot \left(\sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 + \left(v \cdot v\right) \cdot -5}{t \cdot \left(\pi \cdot \left(\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
    Proof
    (/.f64 (+.f64 1 (*.f64 (*.f64 v v) -5)) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 1 (*.f64 (*.f64 v v) (Rewrite<= metadata-eval (neg.f64 5)))) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 1 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 5)))) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 1 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 5 (*.f64 v v))))) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 5 (*.f64 v v)))) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 t (*.f64 (PI.f64) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (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 1 (*.f64 5 (*.f64 v v))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t (PI.f64)) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))))): 42 points increase in error, 25 points decrease in error
    (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (-.f64 1 (*.f64 v v))))): 1 points increase in error, 1 points decrease in error
  3. Final simplification0.4

    \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{t \cdot \left(\pi \cdot \left(\sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]

Alternatives

Alternative 1
Error0.4
Cost14336
\[\frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{t \cdot \pi}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Alternative 2
Error1.1
Cost13184
\[\frac{1}{\left(t \cdot \pi\right) \cdot \sqrt{2}} \]
Alternative 3
Error1.0
Cost13184
\[\frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}} \]
Alternative 4
Error1.3
Cost13056
\[\frac{\sqrt{0.5}}{t \cdot \pi} \]
Alternative 5
Error1.3
Cost13056
\[\frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))