ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.5% → 100.0%

Time: 5.0s

Precision: binary64

Cost: 13120

\[-0.01 \leq x \land x \leq 0.01\]

Math

\[1 - \cos x \]

↓

\[\sin x \cdot \tan \left(\frac{x}{2}\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (* (sin x) (tan (/ x 2.0))))

C

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

↓

double code(double x) {
	return sin(x) * tan((x / 2.0));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin(x) * tan((x / 2.0d0))
end function

Java

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

↓

public static double code(double x) {
	return Math.sin(x) * Math.tan((x / 2.0));
}

Python

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

↓

def code(x):
	return math.sin(x) * math.tan((x / 2.0))

Julia

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

↓

function code(x)
	return Float64(sin(x) * tan(Float64(x / 2.0)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = sin(x) * tan((x / 2.0));
end

Wolfram

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

↓

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	53.5%
Target	100.0%
Herbie	100.0%

\[\frac{\sin x \cdot \sin x}{1 + \cos x} \]

Derivation

1. Initial program 52.3%

   \[1 - \cos x \]

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}} \]
   Step-by-step derivation

   [Start] 52.3%	\[ 1 - \cos x \]
   flip-- [=>] 52.3%	\[ \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}} \]
   div-inv [=>] 52.3%	\[ \color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}} \]
   metadata-eval [=>] 52.3%	\[ \left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x} \]
   1-sub-cos [=>] 100.0%	\[ \color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x} \]
   associate-*r/ [=>] 100.0%	\[ \color{blue}{\frac{\left(\sin x \cdot \sin x\right) \cdot 1}{1 + \cos x}} \]
   *-rgt-identity [=>] 100.0%	\[ \frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x} \]
   associate-*r/ [<=] 100.0%	\[ \color{blue}{\sin x \cdot \frac{\sin x}{1 + \cos x}} \]
   hang-0p-tan [=>] 100.0%	\[ \sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

4. Final simplification 100.0%

   \[\leadsto \sin x \cdot \tan \left(\frac{x}{2}\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13120

\[\sin x \cdot \tan \left(\frac{x}{2}\right) \]

Alternative 2
Accuracy	100.0%
Cost	6976

\[x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\right) \]

Alternative 3
Accuracy	99.9%
Cost	960

\[\left(x \cdot \left(x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right) + 0.5 \cdot \left(x \cdot x\right) \]

Alternative 4
Accuracy	99.6%
Cost	320

\[0.5 \cdot \left(x \cdot x\right) \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :herbie-target
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))