2sin (example 3.3)

Average Error: 37.4 → 0.4

Time: 15.3s

Precision: binary64

Cost: 32576

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (+ (* (cos x) (sin eps)) (- (* (sin x) (cos eps)) (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(x) * 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 = (cos(x) * sin(eps)) + ((sin(x) * 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.cos(x) * Math.sin(eps)) + ((Math.sin(x) * 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.cos(x) * math.sin(eps)) + ((math.sin(x) * 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(cos(x) * sin(eps)) + Float64(Float64(sin(x) * cos(eps)) - 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(x) * 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[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.4
Target	15.4
Herbie	0.4

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

Derivation

1. Initial program 37.4

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

2. Applied egg-rr 0.4

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

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	32448

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

Alternative 2
Error	0.4
Cost	26176

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

Alternative 3
Error	14.6
Cost	26048

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

Alternative 4
Error	15.4
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	15.0
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 6
Error	15.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.4 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	29.1
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	45.3
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023027 
(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)))