2sin (example 3.3)

Average Error: 37.0 → 0.2

Time: 14.9s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double eps) {
	return (cos(x) * sin(eps)) - (sin(eps) * (tan((eps / 2.0)) * 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 = (cos(x) * sin(eps)) - (sin(eps) * (tan((eps / 2.0d0)) * 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.cos(x) * Math.sin(eps)) - (Math.sin(eps) * (Math.tan((eps / 2.0)) * Math.sin(x)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = (cos(x) * sin(eps)) - (sin(eps) * (tan((eps / 2.0)) * 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[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.0
Target	15.1
Herbie	0.2

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

Derivation

1. Initial program 37.0

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

2. Applied egg-rr 21.7

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

3. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)} \]
   Proof
   (fma.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
   (fma.f64 (sin.f64 eps) (cos.f64 x) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (cos.f64 eps) (sin.f64 x)) (*.f64 -1 (sin.f64 x))))): 0 points increase in error, 1 points decrease in error
   (fma.f64 (sin.f64 eps) (cos.f64 x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (cos.f64 eps))) (*.f64 -1 (sin.f64 x)))): 0 points increase in error, 7 points decrease in error
   (fma.f64 (sin.f64 eps) (cos.f64 x) (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x))))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (neg.f64 (sin.f64 x))))): 0 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (sin.f64 eps))) (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (neg.f64 (sin.f64 x)))): 1 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (neg.f64 (sin.f64 x))) (*.f64 (cos.f64 x) (sin.f64 eps)))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (sin.f64 x) (cos.f64 eps)) (+.f64 (neg.f64 (sin.f64 x)) (*.f64 (cos.f64 x) (sin.f64 eps))))): 7 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.4

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

5. Taylor expanded in eps around inf 0.4

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

6. Simplified 0.4

   \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2}}{\cos \varepsilon + 1} \cdot \sin x} \]
   Proof
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (/.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=> *-commutative_binary64 (*.f64 (sin.f64 eps) (cos.f64 x))) (*.f64 (/.f64 (pow.f64 (sin.f64 eps) 2) (+.f64 (cos.f64 eps) 1)) (sin.f64 x))): 9 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (*.f64 (/.f64 (pow.f64 (sin.f64 eps) 2) (Rewrite<= +-commutative_binary64 (+.f64 1 (cos.f64 eps)))) (sin.f64 x))): 0 points increase in error, 9 points decrease in error
   (-.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 (sin.f64 eps) 2) (/.f64 (+.f64 1 (cos.f64 eps)) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 eps) 2) (sin.f64 x)) (+.f64 1 (cos.f64 eps))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2))) (+.f64 1 (cos.f64 eps)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (sin.f64 eps) (cos.f64 x)) (neg.f64 (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)) (+.f64 1 (cos.f64 eps)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (sin.f64 eps))) (neg.f64 (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)) (+.f64 1 (cos.f64 eps))))): 9 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)) (+.f64 1 (cos.f64 eps)))))): 0 points increase in error, 9 points decrease in error

7. Taylor expanded in eps around inf 0.4

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

8. Simplified 0.2

   \[\leadsto \cos x \cdot \sin \varepsilon - \color{blue}{\sin \varepsilon \cdot \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)} \]
   Proof
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (sin.f64 eps) (*.f64 (tan.f64 (/.f64 eps 2)) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (sin.f64 eps) (*.f64 (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps)))) (sin.f64 x)))): 6 points increase in error, 3 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (sin.f64 eps) (*.f64 (/.f64 (sin.f64 eps) (Rewrite=> +-commutative_binary64 (+.f64 (cos.f64 eps) 1))) (sin.f64 x)))): 0 points increase in error, 6 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 eps) (/.f64 (sin.f64 eps) (+.f64 (cos.f64 eps) 1))) (sin.f64 x)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (+.f64 (cos.f64 eps) 1))) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (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 (cos.f64 x) (sin.f64 eps)) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 eps) 2) (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 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2))) (+.f64 (cos.f64 eps) 1))): 9 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (sin.f64 eps)) (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 eps) 2)) (Rewrite<= +-commutative_binary64 (+.f64 1 (cos.f64 eps))))): 0 points increase in error, 9 points decrease in error

9. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	32448

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

Alternative 2
Error	0.4
Cost	26176

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

Alternative 3
Error	14.4
Cost	26048

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

Alternative 4
Error	15.2
Cost	13888

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

Alternative 5
Error	15.2
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00056 \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-27}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 6
Error	15.8
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	28.6
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	61.3
Cost	64

\[0 \]

Alternative 9
Error	45.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022343 
(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)))