Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.3

Time: 10.7s

Precision: binary64

Cost: 14464

Math

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

↓

\[\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(t \cdot \left(-1 + v \cdot v\right)\right)} \]

FPCore

(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)))
  (/ (/ 1.0 PI) (* (sqrt (+ 2.0 (* (* v v) -6.0))) (* t (+ -1.0 (* v v)))))))

C

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))) * ((1.0 / ((double) M_PI)) / (sqrt((2.0 + ((v * v) * -6.0))) * (t * (-1.0 + (v * v)))));
}

Java

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))) * ((1.0 / Math.PI) / (Math.sqrt((2.0 + ((v * v) * -6.0))) * (t * (-1.0 + (v * v)))));
}

Python

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))) * ((1.0 / math.pi) / (math.sqrt((2.0 + ((v * v) * -6.0))) * (t * (-1.0 + (v * v)))))

Julia

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(v * Float64(v * -5.0))) * Float64(Float64(1.0 / pi) / Float64(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * Float64(t * Float64(-1.0 + Float64(v * v))))))
end

MATLAB

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))) * ((1.0 / pi) / (sqrt((2.0 + ((v * v) * -6.0))) * (t * (-1.0 + (v * v)))));
end

Wolfram

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[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / Pi), $MachinePrecision] / N[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * N[(-1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Simplified 0.4

   \[\leadsto \color{blue}{\frac{1 + -5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   Proof
   (/.f64 (+.f64 1 (*.f64 -5 (*.f64 v v))) (*.f64 (*.f64 (PI.f64) t) (*.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 (Rewrite<= metadata-eval (neg.f64 5)) (*.f64 v v))) (*.f64 (*.f64 (PI.f64) t) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 (*.f64 v v) 3)))) (-.f64 1 (*.f64 v v))))): 0 points increase in error, 12 points decrease in error
   (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 5 (*.f64 v v)))) (*.f64 (*.f64 (PI.f64) t) (*.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 (*.f64 (PI.f64) t) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 v v)))))) (-.f64 1 (*.f64 v v))))): 12 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (*.f64 (PI.f64) t) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (*.f64 v v) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (*.f64 (PI.f64) t) (-.f64 (Rewrite=> *-lft-identity_binary64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (*.f64 v v) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (*.f64 (PI.f64) t) (*.f64 (*.f64 v v) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))))))): 0 points increase in error, 12 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))))) (*.f64 (*.f64 (PI.f64) t) (*.f64 (*.f64 v v) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))))))): 12 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (-.f64 (*.f64 1 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))))) (*.f64 (*.f64 (PI.f64) t) (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))) (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (-.f64 (*.f64 1 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.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 v v))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (-.f64 (*.f64 1 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))))) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 v v) (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite=> distribute-rgt-out--_binary64 (*.f64 (*.f64 (*.f64 (PI.f64) t) (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

3. Applied egg-rr 0.4

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

4. Simplified 0.3

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

5. Applied egg-rr 0.3

   \[\leadsto \left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{\frac{1}{\pi}}{\color{blue}{-\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(\left(1 - v \cdot v\right) \cdot t\right)}} \]

6. Final simplification 0.3

   \[\leadsto \left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(t \cdot \left(-1 + v \cdot v\right)\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	14336

\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(\left(1 - v \cdot v\right) \cdot t\right)\right)} \]

Alternative 2
Error	1.1
Cost	13184

\[\frac{1}{\sqrt{2} \cdot \left(\pi \cdot t\right)} \]

Alternative 3
Error	1.1
Cost	13184

\[\frac{\frac{\frac{1}{t}}{\sqrt{2}}}{\pi} \]

Alternative 4
Error	0.8
Cost	13184

\[\frac{\frac{\frac{1}{\sqrt{2}}}{\pi}}{t} \]

Alternative 5
Error	1.4
Cost	13056

\[\frac{\sqrt{0.5}}{\pi \cdot t} \]

Reproduce

herbie shell --seed 2022343 
(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)))))