bug323 (missed optimization)

Average Error: 59.5 → 59.5

Time: 10.6s

Precision: binary64

Cost: 34368

\[0 \leq x \land x \leq 0.5\]

Math

\[\cos^{-1} \left(1 - x\right) \]

↓

\[\begin{array}{l} t_0 := -1 - \cos^{-1} \left(1 - x\right)\\ t_1 := \frac{1}{t_0}\\ \left(0 - t_1 \cdot \left(t_1 \cdot \left(\left(t_0 \cdot t_0\right) \cdot t_0\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- -1.0 (acos (- 1.0 x)))) (t_1 (/ 1.0 t_0)))
   (- (- 0.0 (* t_1 (* t_1 (* (* t_0 t_0) t_0)))) 1.0)))

C

double code(double x) {
	return acos((1.0 - x));
}

↓

double code(double x) {
	double t_0 = -1.0 - acos((1.0 - x));
	double t_1 = 1.0 / t_0;
	return (0.0 - (t_1 * (t_1 * ((t_0 * t_0) * t_0)))) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (-1.0d0) - acos((1.0d0 - x))
    t_1 = 1.0d0 / t_0
    code = (0.0d0 - (t_1 * (t_1 * ((t_0 * t_0) * t_0)))) - 1.0d0
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

↓

public static double code(double x) {
	double t_0 = -1.0 - Math.acos((1.0 - x));
	double t_1 = 1.0 / t_0;
	return (0.0 - (t_1 * (t_1 * ((t_0 * t_0) * t_0)))) - 1.0;
}

Python

def code(x):
	return math.acos((1.0 - x))

↓

def code(x):
	t_0 = -1.0 - math.acos((1.0 - x))
	t_1 = 1.0 / t_0
	return (0.0 - (t_1 * (t_1 * ((t_0 * t_0) * t_0)))) - 1.0

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

↓

function code(x)
	t_0 = Float64(-1.0 - acos(Float64(1.0 - x)))
	t_1 = Float64(1.0 / t_0)
	return Float64(Float64(0.0 - Float64(t_1 * Float64(t_1 * Float64(Float64(t_0 * t_0) * t_0)))) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

↓

function tmp = code(x)
	t_0 = -1.0 - acos((1.0 - x));
	t_1 = 1.0 / t_0;
	tmp = (0.0 - (t_1 * (t_1 * ((t_0 * t_0) * t_0)))) - 1.0;
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(-1.0 - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, N[(N[(0.0 - N[(t$95$1 * N[(t$95$1 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Target

Original	59.5
Target	0.0
Herbie	59.5

\[2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \]

Derivation

1. Initial program 59.5

   \[\cos^{-1} \left(1 - x\right) \]

2. Applied egg-rr 59.5

   \[\leadsto \color{blue}{\left(0 - \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right) - 1} \]

3. Applied egg-rr 59.5

   \[\leadsto \left(0 - \color{blue}{\frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \left(\left(-1 - \cos^{-1} \left(1 - x\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right)}\right) - 1 \]

4. Applied egg-rr 59.5

   \[\leadsto \left(0 - \frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \color{blue}{\left(\left(0 - \left(-1 - \left(-1 - \cos^{-1} \left(1 - x\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right)\right) - 1\right)}\right) - 1 \]

5. Applied egg-rr 59.5

   \[\leadsto \left(0 - \frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \color{blue}{\left(\frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \left(\left(\left(-1 - \cos^{-1} \left(1 - x\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right)\right)}\right) - 1 \]

6. Final simplification 59.5

   \[\leadsto \left(0 - \frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \left(\frac{1}{-1 - \cos^{-1} \left(1 - x\right)} \cdot \left(\left(\left(-1 - \cos^{-1} \left(1 - x\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right) \cdot \left(-1 - \cos^{-1} \left(1 - x\right)\right)\right)\right)\right) - 1 \]

Alternatives

Alternative 1
Error	59.5
Cost	20928

\[\begin{array}{l} t_0 := -1 - \cos^{-1} \left(1 - x\right)\\ \left(0 - \frac{1}{t_0} \cdot \left(\left(1 + t_0 \cdot t_0\right) + -1\right)\right) - 1 \end{array} \]

Alternative 2
Error	59.5
Cost	20672

\[\begin{array}{l} t_0 := -1 - \cos^{-1} \left(1 - x\right)\\ \left(0 - \frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)\right) - 1 \end{array} \]

Alternative 3
Error	59.5
Cost	6848

\[\left(-1 + \cos^{-1} \left(1 - x\right)\right) + 1 \]

Alternative 4
Error	59.5
Cost	6592

\[\cos^{-1} \left(1 - x\right) \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))