Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.6

Time: 16.0s

Precision: binary64

Cost: 51392

Math

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

↓

\[\begin{array}{l} t_0 := v \cdot v + -1\\ t_1 := \cos^{-1} \left(\frac{v \cdot \left(v \cdot -5\right) + 1}{t_0}\right)\\ t_2 := \left(t_1 \cdot \frac{1}{t_1}\right) \cdot t_1\\ \left(t_2 \cdot \frac{1}{\cos^{-1} \left(\frac{\left(v \cdot v\right) \cdot -5 + 1}{t_0}\right)}\right) \cdot t_2 \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (+ (* v v) -1.0))
        (t_1 (acos (/ (+ (* v (* v -5.0)) 1.0) t_0)))
        (t_2 (* (* t_1 (/ 1.0 t_1)) t_1)))
   (* (* t_2 (/ 1.0 (acos (/ (+ (* (* v v) -5.0) 1.0) t_0)))) t_2)))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

double code(double v) {
	double t_0 = (v * v) + -1.0;
	double t_1 = acos((((v * (v * -5.0)) + 1.0) / t_0));
	double t_2 = (t_1 * (1.0 / t_1)) * t_1;
	return (t_2 * (1.0 / acos(((((v * v) * -5.0) + 1.0) / t_0)))) * t_2;
}

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

↓

real(8) function code(v)
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (v * v) + (-1.0d0)
    t_1 = acos((((v * (v * (-5.0d0))) + 1.0d0) / t_0))
    t_2 = (t_1 * (1.0d0 / t_1)) * t_1
    code = (t_2 * (1.0d0 / acos(((((v * v) * (-5.0d0)) + 1.0d0) / t_0)))) * t_2
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

public static double code(double v) {
	double t_0 = (v * v) + -1.0;
	double t_1 = Math.acos((((v * (v * -5.0)) + 1.0) / t_0));
	double t_2 = (t_1 * (1.0 / t_1)) * t_1;
	return (t_2 * (1.0 / Math.acos(((((v * v) * -5.0) + 1.0) / t_0)))) * t_2;
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

↓

def code(v):
	t_0 = (v * v) + -1.0
	t_1 = math.acos((((v * (v * -5.0)) + 1.0) / t_0))
	t_2 = (t_1 * (1.0 / t_1)) * t_1
	return (t_2 * (1.0 / math.acos(((((v * v) * -5.0) + 1.0) / t_0)))) * t_2

Julia

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)
	t_0 = Float64(Float64(v * v) + -1.0)
	t_1 = acos(Float64(Float64(Float64(v * Float64(v * -5.0)) + 1.0) / t_0))
	t_2 = Float64(Float64(t_1 * Float64(1.0 / t_1)) * t_1)
	return Float64(Float64(t_2 * Float64(1.0 / acos(Float64(Float64(Float64(Float64(v * v) * -5.0) + 1.0) / t_0)))) * t_2)
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

↓

function tmp = code(v)
	t_0 = (v * v) + -1.0;
	t_1 = acos((((v * (v * -5.0)) + 1.0) / t_0));
	t_2 = (t_1 * (1.0 / t_1)) * t_1;
	tmp = (t_2 * (1.0 / acos(((((v * v) * -5.0) + 1.0) / t_0)))) * t_2;
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]

↓

code[v_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(N[(N[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[(t$95$2 * N[(1.0 / N[ArcCos[N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]

Derivation

1. Initial program 0.6

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

2. Applied egg-rr 0.6

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

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.6

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

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	36672

\[\begin{array}{l} t_0 := v \cdot v + -1\\ t_1 := \cos^{-1} \left(\frac{\left(v \cdot v\right) \cdot -5 + 1}{t_0}\right)\\ t_2 := \cos^{-1} \left(\frac{v \cdot \left(v \cdot -5\right) + 1}{t_0}\right)\\ \left(\left(\left(t_2 \cdot \frac{1}{t_2}\right) \cdot t_2\right) \cdot \frac{1}{t_1}\right) \cdot t_1 \end{array} \]

Alternative 2
Error	0.6
Cost	21952

\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{\left(v \cdot v\right) \cdot -5 + 1}{v \cdot v + -1}\right)\\ \left(t_0 \cdot \frac{1}{t_0}\right) \cdot t_0 \end{array} \]

Alternative 3
Error	0.6
Cost	7232

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

Alternative 4
Error	1.3
Cost	6464

\[\cos^{-1} -1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))