2sin (example 3.3)

Average Error: 37.0 → 0.3

Time: 12.4s

Precision: binary64

Cost: 39168

Math

\[\sin \left(x + \varepsilon\right) - \sin x \]

↓

\[\sin \varepsilon \cdot \cos x + \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} \cdot \sin x \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (+
  (* (sin eps) (cos x))
  (* (/ (pow (sin eps) 2.0) (- -1.0 (cos eps))) (sin x))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	return (sin(eps) * cos(x)) + ((pow(sin(eps), 2.0) / (-1.0 - cos(eps))) * sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) + (((sin(eps) ** 2.0d0) / ((-1.0d0) - cos(eps))) * sin(x))
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

↓

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) + ((Math.pow(Math.sin(eps), 2.0) / (-1.0 - Math.cos(eps))) * Math.sin(x));
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

↓

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) + ((math.pow(math.sin(eps), 2.0) / (-1.0 - math.cos(eps))) * math.sin(x))

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

↓

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) + Float64(Float64((sin(eps) ^ 2.0) / Float64(-1.0 - cos(eps))) * sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

↓

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) + (((sin(eps) ^ 2.0) / (-1.0 - cos(eps))) * sin(x));
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.0
Target	15.0
Herbie	0.3

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Initial program 37.0

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Applied egg-rr 21.9

   \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
   Proof
   (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (*.f64 (sin.f64 x) (+.f64 (cos.f64 eps) -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (sin.f64 eps))) (*.f64 (sin.f64 x) (+.f64 (cos.f64 eps) -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (sin.f64 x) (Rewrite=> +-commutative_binary64 (+.f64 -1 (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (sin.f64 x)) (*.f64 (cos.f64 eps) (sin.f64 x))))): 11 points increase in error, 7 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x))) (*.f64 (cos.f64 eps) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (+.f64 (neg.f64 (sin.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (neg.f64 (sin.f64 x))) (*.f64 (sin.f64 x) (cos.f64 eps)))): 110 points increase in error, 12 points decrease in error
   (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (sin.f64 x))) (*.f64 (sin.f64 x) (cos.f64 eps))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.3

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{\left(-{\sin \varepsilon}^{2}\right) \cdot \sin x}{\cos \varepsilon + 1}} \]

5. Simplified 0.3

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} \cdot \sin x} \]
   Proof
   (*.f64 (/.f64 (pow.f64 (sin.f64 eps) 2) (-.f64 -1 (cos.f64 eps))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)))) (-.f64 -1 (cos.f64 eps))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2))) (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 (cos.f64 eps))))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2))) (+.f64 (Rewrite<= metadata-eval (neg.f64 1)) (neg.f64 (cos.f64 eps)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 1 (cos.f64 eps))))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2))) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (neg.f64 (pow.f64 (sin.f64 eps) 2)))) (neg.f64 (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (*.f64 -1 (neg.f64 (pow.f64 (sin.f64 eps) 2))) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1)))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite=> *-lft-identity_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (/.f64 (+.f64 (cos.f64 eps) 1) (sin.f64 x)))): 20 points increase in error, 32 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (sin.f64 x)) (+.f64 (cos.f64 eps) 1))): 36 points increase in error, 26 points decrease in error

6. Final simplification 0.3

   \[\leadsto \sin \varepsilon \cdot \cos x + \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} \cdot \sin x \]

Alternatives

Alternative 1
Error	0.4
Cost	38848

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x - \sin x\right) \]

Alternative 2
Error	0.4
Cost	32448

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(-1 + \cos \varepsilon\right)\right) \]

Alternative 3
Error	0.4
Cost	26304

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

Alternative 4
Error	0.4
Cost	26176

\[\sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right) \]

Alternative 5
Error	14.5
Cost	13768

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.29:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot -0.5\right) + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.5
Cost	13640

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.29:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.0
Cost	13632

\[\cos \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 8
Error	15.0
Cost	13632

\[\left(2 \cdot \cos \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) \]

Alternative 9
Error	14.6
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.29:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.0
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 11
Error	28.4
Cost	6464

\[\sin \varepsilon \]

Alternative 12
Error	44.7
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))